memgraph/tests/benchmark/btree.hpp

/*******************************************************************************
 * tlx/container/btree.hpp
 *
 * Part of tlx - http://panthema.net/tlx
 *
 * Copyright (C) 2008-2017 Timo Bingmann <tb@panthema.net>
 *
 * All rights reserved. Published under the Boost Software License, Version 1.0
 ******************************************************************************/

#ifndef TLX_CONTAINER_BTREE_HEADER
#define TLX_CONTAINER_BTREE_HEADER

#include "core.hpp"

// *** Required Headers from the STL

#include <algorithm>
#include <cassert>
#include <cstddef>
#include <functional>
#include <istream>
#include <memory>
#include <ostream>
#include <utility>

namespace tlx {

//! \addtogroup tlx_container
//! \{
//! \defgroup tlx_container_btree B+ Trees
//! B+ tree variants
//! \{

// *** Debugging Macros

#ifdef TLX_BTREE_DEBUG

#include <iostream>

//! Print out debug information to std::cout if TLX_BTREE_DEBUG is defined.
#define TLX_BTREE_PRINT(x)                    \
  do {                                        \
    if (debug) (std::cout << x << std::endl); \
  } while (0)

//! Assertion only if TLX_BTREE_DEBUG is defined. This is not used in verify().
#define TLX_BTREE_ASSERT(x) \
  do {                      \
    assert(x);              \
  } while (0)

#else

//! Print out debug information to std::cout if TLX_BTREE_DEBUG is defined.
#define TLX_BTREE_PRINT(x) \
  do {                     \
  } while (0)

//! Assertion only if TLX_BTREE_DEBUG is defined. This is not used in verify().
#define TLX_BTREE_ASSERT(x) \
  do {                      \
  } while (0)

#endif

//! The maximum of a and b. Used in some compile-time formulas.
#define TLX_BTREE_MAX(a, b) ((a) < (b) ? (b) : (a))

#ifndef TLX_BTREE_FRIENDS
//! The macro TLX_BTREE_FRIENDS can be used by outside class to access the B+
//! tree internals. This was added for wxBTreeDemo to be able to draw the
//! tree.
#define TLX_BTREE_FRIENDS friend class btree_friend
#endif

/*!
 * Generates default traits for a B+ tree used as a set or map. It estimates
 * leaf and inner node sizes by assuming a cache line multiple of 256 bytes.
 */
template <typename Key, typename Value>
struct btree_default_traits {
  //! If true, the tree will self verify its invariants after each insert() or
  //! erase(). The header must have been compiled with TLX_BTREE_DEBUG
  //! defined.
  static const bool self_verify = false;

  //! If true, the tree will print out debug information and a tree dump
  //! during insert() or erase() operation. The header must have been
  //! compiled with TLX_BTREE_DEBUG defined and key_type must be std::ostream
  //! printable.
  static const bool debug = false;

  //! Number of slots in each leaf of the tree. Estimated so that each node
  //! has a size of about 256 bytes.
  static const int leaf_slots = TLX_BTREE_MAX(8, 256 / (sizeof(Value)));

  //! Number of slots in each inner node of the tree. Estimated so that each
  //! node has a size of about 256 bytes.
  static const int inner_slots = TLX_BTREE_MAX(8, 256 / (sizeof(Key) + sizeof(void *)));

  //! As of stx-btree-0.9, the code does linear search in find_lower() and
  //! find_upper() instead of binary_search, unless the node size is larger
  //! than this threshold. See notes at
  //! http://panthema.net/2013/0504-STX-B+Tree-Binary-vs-Linear-Search
  static const size_t binsearch_threshold = 256;
};

/*!
 * Basic class implementing a B+ tree data structure in memory.
 *
 * The base implementation of an in-memory B+ tree. It is based on the
 * implementation in Cormen's Introduction into Algorithms, Jan Jannink's paper
 * and other algorithm resources. Almost all STL-required function calls are
 * implemented. The asymptotic time requirements of the STL are not always
 * fulfilled in theory, however, in practice this B+ tree performs better than a
 * red-black tree and almost always uses less memory. The insertion function
 * splits the nodes on the recursion unroll. Erase is largely based on Jannink's
 * ideas.
 *
 * This class is specialized into btree_set, btree_multiset, btree_map and
 * btree_multimap using default template parameters and facade functions.
 */
template <typename Key, typename Value, typename KeyOfValue, typename Compare = std::less<Key>,
          typename Traits = btree_default_traits<Key, Value>, bool Duplicates = false,
          typename Allocator = std::allocator<Value>>
class BTree {
 public:
  //! \name Template Parameter Types
  //! \{

  //! First template parameter: The key type of the B+ tree. This is stored in
  //! inner nodes.
  typedef Key key_type;

  //! Second template parameter: Composition pair of key and data types, or
  //! just the key for set containers. This data type is stored in the leaves.
  typedef Value value_type;

  //! Third template: key extractor class to pull key_type from value_type.
  typedef KeyOfValue key_of_value;

  //! Fourth template parameter: key_type comparison function object
  typedef Compare key_compare;

  //! Fifth template parameter: Traits object used to define more parameters
  //! of the B+ tree
  typedef Traits traits;

  //! Sixth template parameter: Allow duplicate keys in the B+ tree. Used to
  //! implement multiset and multimap.
  static const bool allow_duplicates = Duplicates;

  //! Seventh template parameter: STL allocator for tree nodes
  typedef Allocator allocator_type;

  //! \}

  // The macro TLX_BTREE_FRIENDS can be used by outside class to access the B+
  // tree internals. This was added for wxBTreeDemo to be able to draw the
  // tree.
  TLX_BTREE_FRIENDS;

 public:
  //! \name Constructed Types
  //! \{

  //! Typedef of our own type
  typedef BTree<key_type, value_type, key_of_value, key_compare, traits, allow_duplicates, allocator_type> Self;

  //! Size type used to count keys
  typedef size_t size_type;

  //! \}

 public:
  //! \name Static Constant Options and Values of the B+ Tree
  //! \{

  //! Base B+ tree parameter: The number of key/data slots in each leaf
  static const unsigned short leaf_slotmax = traits::leaf_slots;

  //! Base B+ tree parameter: The number of key slots in each inner node,
  //! this can differ from slots in each leaf.
  static const unsigned short inner_slotmax = traits::inner_slots;

  //! Computed B+ tree parameter: The minimum number of key/data slots used
  //! in a leaf. If fewer slots are used, the leaf will be merged or slots
  //! shifted from it's siblings.
  static const unsigned short leaf_slotmin = (leaf_slotmax / 2);

  //! Computed B+ tree parameter: The minimum number of key slots used
  //! in an inner node. If fewer slots are used, the inner node will be
  //! merged or slots shifted from it's siblings.
  static const unsigned short inner_slotmin = (inner_slotmax / 2);

  //! Debug parameter: Enables expensive and thorough checking of the B+ tree
  //! invariants after each insert/erase operation.
  static const bool self_verify = traits::self_verify;

  //! Debug parameter: Prints out lots of debug information about how the
  //! algorithms change the tree. Requires the header file to be compiled
  //! with TLX_BTREE_DEBUG and the key type must be std::ostream printable.
  static const bool debug = traits::debug;

  //! \}

 private:
  //! \name Node Classes for In-Memory Nodes
  //! \{

  //! The header structure of each node in-memory. This structure is extended
  //! by InnerNode or LeafNode.
  struct node {
    //! Level in the b-tree, if level == 0 -> leaf node
    unsigned short level;

    //! Number of key slotuse use, so the number of valid children or data
    //! pointers
    unsigned short slotuse;

    //! Delayed initialisation of constructed node.
    void initialize(const unsigned short l) {
      level = l;
      slotuse = 0;
    }

    //! True if this is a leaf node.
    bool is_leafnode() const { return (level == 0); }
  };

  //! Extended structure of a inner node in-memory. Contains only keys and no
  //! data items.
  struct InnerNode : public node {
    //! Define an related allocator for the InnerNode structs.
    typedef typename std::allocator_traits<Allocator>::template rebind_alloc<InnerNode> alloc_type;

    //! Keys of children or data pointers
    key_type slotkey[inner_slotmax];  // NOLINT

    //! Pointers to children
    node *childid[inner_slotmax + 1];  // NOLINT

    //! Set variables to initial values.
    void initialize(const unsigned short l) { node::initialize(l); }

    //! Return key in slot s
    const key_type &key(size_t s) const { return slotkey[s]; }

    //! True if the node's slots are full.
    bool is_full() const { return (node::slotuse == inner_slotmax); }

    //! True if few used entries, less than half full.
    bool is_few() const { return (node::slotuse <= inner_slotmin); }

    //! True if node has too few entries.
    bool is_underflow() const { return (node::slotuse < inner_slotmin); }
  };

  //! Extended structure of a leaf node in memory. Contains pairs of keys and
  //! data items. Key and data slots are kept together in value_type.
  struct LeafNode : public node {
    //! Define an related allocator for the LeafNode structs.
    typedef typename std::allocator_traits<Allocator>::template rebind_alloc<LeafNode> alloc_type;

    //! Double linked list pointers to traverse the leaves
    LeafNode *prev_leaf;

    //! Double linked list pointers to traverse the leaves
    LeafNode *next_leaf;

    //! Array of (key, data) pairs
    value_type slotdata[leaf_slotmax];  // NOLINT

    //! Set variables to initial values
    void initialize() {
      node::initialize(0);
      prev_leaf = next_leaf = nullptr;
    }

    //! Return key in slot s.
    const key_type &key(size_t s) const { return key_of_value::get(slotdata[s]); }

    //! True if the node's slots are full.
    bool is_full() const { return (node::slotuse == leaf_slotmax); }

    //! True if few used entries, less than half full.
    bool is_few() const { return (node::slotuse <= leaf_slotmin); }

    //! True if node has too few entries.
    bool is_underflow() const { return (node::slotuse < leaf_slotmin); }

    //! Set the (key,data) pair in slot. Overloaded function used by
    //! bulk_load().
    void set_slot(unsigned short slot, const value_type &value) {
      TLX_BTREE_ASSERT(slot < node::slotuse);
      slotdata[slot] = value;
    }
  };

  //! \}

 public:
  //! \name Iterators and Reverse Iterators
  //! \{

  class iterator;
  class const_iterator;
  class reverse_iterator;
  class const_reverse_iterator;

  //! STL-like iterator object for B+ tree items. The iterator points to a
  //! specific slot number in a leaf.
  class iterator {
   public:
    // *** Types

    //! The key type of the btree. Returned by key().
    typedef typename BTree::key_type key_type;

    //! The value type of the btree. Returned by operator*().
    typedef typename BTree::value_type value_type;

    //! Reference to the value_type. STL required.
    typedef value_type &reference;

    //! Pointer to the value_type. STL required.
    typedef value_type *pointer;

    //! STL-magic iterator category
    typedef std::bidirectional_iterator_tag iterator_category;

    //! STL-magic
    typedef ptrdiff_t difference_type;

    //! Our own type
    typedef iterator self;

   private:
    // *** Members

    //! The currently referenced leaf node of the tree
    typename BTree::LeafNode *curr_leaf;

    //! Current key/data slot referenced
    unsigned short curr_slot;

    //! Friendly to the const_iterator, so it may access the two data items
    //! directly.
    friend class const_iterator;

    //! Also friendly to the reverse_iterator, so it may access the two
    //! data items directly.
    friend class reverse_iterator;

    //! Also friendly to the const_reverse_iterator, so it may access the
    //! two data items directly.
    friend class const_reverse_iterator;

    //! Also friendly to the base btree class, because erase_iter() needs
    //! to read the curr_leaf and curr_slot values directly.
    friend class BTree<key_type, value_type, key_of_value, key_compare, traits, allow_duplicates, allocator_type>;

    // The macro TLX_BTREE_FRIENDS can be used by outside class to access
    // the B+ tree internals. This was added for wxBTreeDemo to be able to
    // draw the tree.
    TLX_BTREE_FRIENDS;

   public:
    // *** Methods

    //! Default-Constructor of a mutable iterator
    iterator() : curr_leaf(nullptr), curr_slot(0) {}

    //! Initializing-Constructor of a mutable iterator
    iterator(typename BTree::LeafNode *l, unsigned short s) : curr_leaf(l), curr_slot(s) {}

    //! Copy-constructor from a reverse iterator
    iterator(const reverse_iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Dereference the iterator.
    reference operator*() const { return curr_leaf->slotdata[curr_slot]; }

    //! Dereference the iterator.
    pointer operator->() const { return &curr_leaf->slotdata[curr_slot]; }

    //! Key of the current slot.
    const key_type &key() const { return curr_leaf->key(curr_slot); }

    //! Prefix++ advance the iterator to the next slot.
    iterator &operator++() {
      if (curr_slot + 1u < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 0;
      } else {
        // this is end()
        curr_slot = curr_leaf->slotuse;
      }

      return *this;
    }

    //! Postfix++ advance the iterator to the next slot.
    iterator operator++(int) {
      iterator tmp = *this;  // copy ourselves

      if (curr_slot + 1u < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 0;
      } else {
        // this is end()
        curr_slot = curr_leaf->slotuse;
      }

      return tmp;
    }

    //! Prefix-- backstep the iterator to the last slot.
    iterator &operator--() {
      if (curr_slot > 0) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse - 1;
      } else {
        // this is begin()
        curr_slot = 0;
      }

      return *this;
    }

    //! Postfix-- backstep the iterator to the last slot.
    iterator operator--(int) {
      iterator tmp = *this;  // copy ourselves

      if (curr_slot > 0) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse - 1;
      } else {
        // this is begin()
        curr_slot = 0;
      }

      return tmp;
    }

    //! Equality of iterators.
    bool operator==(const iterator &x) const { return (x.curr_leaf == curr_leaf) && (x.curr_slot == curr_slot); }

    //! Inequality of iterators.
    bool operator!=(const iterator &x) const { return (x.curr_leaf != curr_leaf) || (x.curr_slot != curr_slot); }
  };

  //! STL-like read-only iterator object for B+ tree items. The iterator
  //! points to a specific slot number in a leaf.
  class const_iterator {
   public:
    // *** Types

    //! The key type of the btree. Returned by key().
    typedef typename BTree::key_type key_type;

    //! The value type of the btree. Returned by operator*().
    typedef typename BTree::value_type value_type;

    //! Reference to the value_type. STL required.
    typedef const value_type &reference;

    //! Pointer to the value_type. STL required.
    typedef const value_type *pointer;

    //! STL-magic iterator category
    typedef std::bidirectional_iterator_tag iterator_category;

    //! STL-magic
    typedef ptrdiff_t difference_type;

    //! Our own type
    typedef const_iterator self;

   private:
    // *** Members

    //! The currently referenced leaf node of the tree
    const typename BTree::LeafNode *curr_leaf;

    //! Current key/data slot referenced
    unsigned short curr_slot;

    //! Friendly to the reverse_const_iterator, so it may access the two
    //! data items directly
    friend class const_reverse_iterator;

    // The macro TLX_BTREE_FRIENDS can be used by outside class to access
    // the B+ tree internals. This was added for wxBTreeDemo to be able to
    // draw the tree.
    TLX_BTREE_FRIENDS;

   public:
    // *** Methods

    //! Default-Constructor of a const iterator
    const_iterator() : curr_leaf(nullptr), curr_slot(0) {}

    //! Initializing-Constructor of a const iterator
    const_iterator(const typename BTree::LeafNode *l, unsigned short s) : curr_leaf(l), curr_slot(s) {}

    //! Copy-constructor from a mutable iterator
    const_iterator(const iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Copy-constructor from a mutable reverse iterator
    const_iterator(const reverse_iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Copy-constructor from a const reverse iterator
    const_iterator(const const_reverse_iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Dereference the iterator.
    reference operator*() const { return curr_leaf->slotdata[curr_slot]; }

    //! Dereference the iterator.
    pointer operator->() const { return &curr_leaf->slotdata[curr_slot]; }

    //! Key of the current slot.
    const key_type &key() const { return curr_leaf->key(curr_slot); }

    //! Prefix++ advance the iterator to the next slot.
    const_iterator &operator++() {
      if (curr_slot + 1u < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 0;
      } else {
        // this is end()
        curr_slot = curr_leaf->slotuse;
      }

      return *this;
    }

    //! Postfix++ advance the iterator to the next slot.
    const_iterator operator++(int) {
      const_iterator tmp = *this;  // copy ourselves

      if (curr_slot + 1u < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 0;
      } else {
        // this is end()
        curr_slot = curr_leaf->slotuse;
      }

      return tmp;
    }

    //! Prefix-- backstep the iterator to the last slot.
    const_iterator &operator--() {
      if (curr_slot > 0) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse - 1;
      } else {
        // this is begin()
        curr_slot = 0;
      }

      return *this;
    }

    //! Postfix-- backstep the iterator to the last slot.
    const_iterator operator--(int) {
      const_iterator tmp = *this;  // copy ourselves

      if (curr_slot > 0) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse - 1;
      } else {
        // this is begin()
        curr_slot = 0;
      }

      return tmp;
    }

    //! Equality of iterators.
    bool operator==(const const_iterator &x) const { return (x.curr_leaf == curr_leaf) && (x.curr_slot == curr_slot); }

    //! Inequality of iterators.
    bool operator!=(const const_iterator &x) const { return (x.curr_leaf != curr_leaf) || (x.curr_slot != curr_slot); }
  };

  //! STL-like mutable reverse iterator object for B+ tree items. The
  //! iterator points to a specific slot number in a leaf.
  class reverse_iterator {
   public:
    // *** Types

    //! The key type of the btree. Returned by key().
    typedef typename BTree::key_type key_type;

    //! The value type of the btree. Returned by operator*().
    typedef typename BTree::value_type value_type;

    //! Reference to the value_type. STL required.
    typedef value_type &reference;

    //! Pointer to the value_type. STL required.
    typedef value_type *pointer;

    //! STL-magic iterator category
    typedef std::bidirectional_iterator_tag iterator_category;

    //! STL-magic
    typedef ptrdiff_t difference_type;

    //! Our own type
    typedef reverse_iterator self;

   private:
    // *** Members

    //! The currently referenced leaf node of the tree
    typename BTree::LeafNode *curr_leaf;

    //! One slot past the current key/data slot referenced.
    unsigned short curr_slot;

    //! Friendly to the const_iterator, so it may access the two data items
    //! directly
    friend class iterator;

    //! Also friendly to the const_iterator, so it may access the two data
    //! items directly
    friend class const_iterator;

    //! Also friendly to the const_iterator, so it may access the two data
    //! items directly
    friend class const_reverse_iterator;

    // The macro TLX_BTREE_FRIENDS can be used by outside class to access
    // the B+ tree internals. This was added for wxBTreeDemo to be able to
    // draw the tree.
    TLX_BTREE_FRIENDS;

   public:
    // *** Methods

    //! Default-Constructor of a reverse iterator
    reverse_iterator() : curr_leaf(nullptr), curr_slot(0) {}

    //! Initializing-Constructor of a mutable reverse iterator
    reverse_iterator(typename BTree::LeafNode *l, unsigned short s) : curr_leaf(l), curr_slot(s) {}

    //! Copy-constructor from a mutable iterator
    reverse_iterator(const iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Dereference the iterator.
    reference operator*() const {
      TLX_BTREE_ASSERT(curr_slot > 0);
      return curr_leaf->slotdata[curr_slot - 1];
    }

    //! Dereference the iterator.
    pointer operator->() const {
      TLX_BTREE_ASSERT(curr_slot > 0);
      return &curr_leaf->slotdata[curr_slot - 1];
    }

    //! Key of the current slot.
    const key_type &key() const {
      TLX_BTREE_ASSERT(curr_slot > 0);
      return curr_leaf->key(curr_slot - 1);
    }

    //! Prefix++ advance the iterator to the next slot.
    reverse_iterator &operator++() {
      if (curr_slot > 1) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse;
      } else {
        // this is begin() == rend()
        curr_slot = 0;
      }

      return *this;
    }

    //! Postfix++ advance the iterator to the next slot.
    reverse_iterator operator++(int) {
      reverse_iterator tmp = *this;  // copy ourselves

      if (curr_slot > 1) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse;
      } else {
        // this is begin() == rend()
        curr_slot = 0;
      }

      return tmp;
    }

    //! Prefix-- backstep the iterator to the last slot.
    reverse_iterator &operator--() {
      if (curr_slot < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 1;
      } else {
        // this is end() == rbegin()
        curr_slot = curr_leaf->slotuse;
      }

      return *this;
    }

    //! Postfix-- backstep the iterator to the last slot.
    reverse_iterator operator--(int) {
      reverse_iterator tmp = *this;  // copy ourselves

      if (curr_slot < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 1;
      } else {
        // this is end() == rbegin()
        curr_slot = curr_leaf->slotuse;
      }

      return tmp;
    }

    //! Equality of iterators.
    bool operator==(const reverse_iterator &x) const {
      return (x.curr_leaf == curr_leaf) && (x.curr_slot == curr_slot);
    }

    //! Inequality of iterators.
    bool operator!=(const reverse_iterator &x) const {
      return (x.curr_leaf != curr_leaf) || (x.curr_slot != curr_slot);
    }
  };

  //! STL-like read-only reverse iterator object for B+ tree items. The
  //! iterator points to a specific slot number in a leaf.
  class const_reverse_iterator {
   public:
    // *** Types

    //! The key type of the btree. Returned by key().
    typedef typename BTree::key_type key_type;

    //! The value type of the btree. Returned by operator*().
    typedef typename BTree::value_type value_type;

    //! Reference to the value_type. STL required.
    typedef const value_type &reference;

    //! Pointer to the value_type. STL required.
    typedef const value_type *pointer;

    //! STL-magic iterator category
    typedef std::bidirectional_iterator_tag iterator_category;

    //! STL-magic
    typedef ptrdiff_t difference_type;

    //! Our own type
    typedef const_reverse_iterator self;

   private:
    // *** Members

    //! The currently referenced leaf node of the tree
    const typename BTree::LeafNode *curr_leaf;

    //! One slot past the current key/data slot referenced.
    unsigned short curr_slot;

    //! Friendly to the const_iterator, so it may access the two data items
    //! directly.
    friend class reverse_iterator;

    // The macro TLX_BTREE_FRIENDS can be used by outside class to access
    // the B+ tree internals. This was added for wxBTreeDemo to be able to
    // draw the tree.
    TLX_BTREE_FRIENDS;

   public:
    // *** Methods

    //! Default-Constructor of a const reverse iterator.
    const_reverse_iterator() : curr_leaf(nullptr), curr_slot(0) {}

    //! Initializing-Constructor of a const reverse iterator.
    const_reverse_iterator(const typename BTree::LeafNode *l, unsigned short s) : curr_leaf(l), curr_slot(s) {}

    //! Copy-constructor from a mutable iterator.
    const_reverse_iterator(const iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Copy-constructor from a const iterator.
    const_reverse_iterator(const const_iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Copy-constructor from a mutable reverse iterator.
    const_reverse_iterator(const reverse_iterator &it)  // NOLINT
        : curr_leaf(it.curr_leaf), curr_slot(it.curr_slot) {}

    //! Dereference the iterator.
    reference operator*() const {
      TLX_BTREE_ASSERT(curr_slot > 0);
      return curr_leaf->slotdata[curr_slot - 1];
    }

    //! Dereference the iterator.
    pointer operator->() const {
      TLX_BTREE_ASSERT(curr_slot > 0);
      return &curr_leaf->slotdata[curr_slot - 1];
    }

    //! Key of the current slot.
    const key_type &key() const {
      TLX_BTREE_ASSERT(curr_slot > 0);
      return curr_leaf->key(curr_slot - 1);
    }

    //! Prefix++ advance the iterator to the previous slot.
    const_reverse_iterator &operator++() {
      if (curr_slot > 1) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse;
      } else {
        // this is begin() == rend()
        curr_slot = 0;
      }

      return *this;
    }

    //! Postfix++ advance the iterator to the previous slot.
    const_reverse_iterator operator++(int) {
      const_reverse_iterator tmp = *this;  // copy ourselves

      if (curr_slot > 1) {
        --curr_slot;
      } else if (curr_leaf->prev_leaf != nullptr) {
        curr_leaf = curr_leaf->prev_leaf;
        curr_slot = curr_leaf->slotuse;
      } else {
        // this is begin() == rend()
        curr_slot = 0;
      }

      return tmp;
    }

    //! Prefix-- backstep the iterator to the next slot.
    const_reverse_iterator &operator--() {
      if (curr_slot < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 1;
      } else {
        // this is end() == rbegin()
        curr_slot = curr_leaf->slotuse;
      }

      return *this;
    }

    //! Postfix-- backstep the iterator to the next slot.
    const_reverse_iterator operator--(int) {
      const_reverse_iterator tmp = *this;  // copy ourselves

      if (curr_slot < curr_leaf->slotuse) {
        ++curr_slot;
      } else if (curr_leaf->next_leaf != nullptr) {
        curr_leaf = curr_leaf->next_leaf;
        curr_slot = 1;
      } else {
        // this is end() == rbegin()
        curr_slot = curr_leaf->slotuse;
      }

      return tmp;
    }

    //! Equality of iterators.
    bool operator==(const const_reverse_iterator &x) const {
      return (x.curr_leaf == curr_leaf) && (x.curr_slot == curr_slot);
    }

    //! Inequality of iterators.
    bool operator!=(const const_reverse_iterator &x) const {
      return (x.curr_leaf != curr_leaf) || (x.curr_slot != curr_slot);
    }
  };

  //! \}

 public:
  //! \name Small Statistics Structure
  //! \{

  /*!
   * A small struct containing basic statistics about the B+ tree. It can be
   * fetched using get_stats().
   */
  struct tree_stats {
    //! Number of items in the B+ tree
    size_type size;

    //! Number of leaves in the B+ tree
    size_type leaves;

    //! Number of inner nodes in the B+ tree
    size_type inner_nodes;

    //! Base B+ tree parameter: The number of key/data slots in each leaf
    static const unsigned short leaf_slots = Self::leaf_slotmax;

    //! Base B+ tree parameter: The number of key slots in each inner node.
    static const unsigned short inner_slots = Self::inner_slotmax;

    //! Zero initialized
    tree_stats() : size(0), leaves(0), inner_nodes(0) {}

    //! Return the total number of nodes
    size_type nodes() const { return inner_nodes + leaves; }

    //! Return the average fill of leaves
    double avgfill_leaves() const { return static_cast<double>(size) / (leaves * leaf_slots); }
  };

  //! \}

 private:
  //! \name Tree Object Data Members
  //! \{

  //! Pointer to the B+ tree's root node, either leaf or inner node.
  node *root_;

  //! Pointer to first leaf in the double linked leaf chain.
  LeafNode *head_leaf_;

  //! Pointer to last leaf in the double linked leaf chain.
  LeafNode *tail_leaf_;

  //! Other small statistics about the B+ tree.
  tree_stats stats_;

  //! Key comparison object. More comparison functions are generated from
  //! this < relation.
  key_compare key_less_;

  //! Memory allocator.
  allocator_type allocator_;

  //! \}

 public:
  //! \name Constructors and Destructor
  //! \{

  //! Default constructor initializing an empty B+ tree with the standard key
  //! comparison function.
  explicit BTree(const allocator_type &alloc = allocator_type())
      : root_(nullptr), head_leaf_(nullptr), tail_leaf_(nullptr), allocator_(alloc) {}

  //! Constructor initializing an empty B+ tree with a special key
  //! comparison object.
  explicit BTree(const key_compare &kcf, const allocator_type &alloc = allocator_type())
      : root_(nullptr), head_leaf_(nullptr), tail_leaf_(nullptr), key_less_(kcf), allocator_(alloc) {}

  //! Constructor initializing a B+ tree with the range [first,last). The
  //! range need not be sorted. To create a B+ tree from a sorted range, use
  //! bulk_load().
  template <class InputIterator>
  BTree(InputIterator first, InputIterator last, const allocator_type &alloc = allocator_type())
      : root_(nullptr), head_leaf_(nullptr), tail_leaf_(nullptr), allocator_(alloc) {
    insert(first, last);
  }

  //! Constructor initializing a B+ tree with the range [first,last) and a
  //! special key comparison object.  The range need not be sorted. To create
  //! a B+ tree from a sorted range, use bulk_load().
  template <class InputIterator>
  BTree(InputIterator first, InputIterator last, const key_compare &kcf, const allocator_type &alloc = allocator_type())
      : root_(nullptr), head_leaf_(nullptr), tail_leaf_(nullptr), key_less_(kcf), allocator_(alloc) {
    insert(first, last);
  }

  //! Frees up all used B+ tree memory pages
  ~BTree() { clear(); }

  //! Fast swapping of two identical B+ tree objects.
  void swap(BTree &from) {
    std::swap(root_, from.root_);
    std::swap(head_leaf_, from.head_leaf_);
    std::swap(tail_leaf_, from.tail_leaf_);
    std::swap(stats_, from.stats_);
    std::swap(key_less_, from.key_less_);
    std::swap(allocator_, from.allocator_);
  }

  //! \}

 public:
  //! \name Key and Value Comparison Function Objects
  //! \{

  //! Function class to compare value_type objects. Required by the STL
  class value_compare {
   protected:
    //! Key comparison function from the template parameter
    key_compare key_comp;

    //! Constructor called from BTree::value_comp()
    explicit value_compare(key_compare kc) : key_comp(kc) {}

    //! Friendly to the btree class so it may call the constructor
    friend class BTree<key_type, value_type, key_of_value, key_compare, traits, allow_duplicates, allocator_type>;

   public:
    //! Function call "less"-operator resulting in true if x < y.
    bool operator()(const value_type &x, const value_type &y) const { return key_comp(x.first, y.first); }
  };

  //! Constant access to the key comparison object sorting the B+ tree.
  key_compare key_comp() const { return key_less_; }

  //! Constant access to a constructed value_type comparison object. Required
  //! by the STL.
  value_compare value_comp() const { return value_compare(key_less_); }

  //! \}

 private:
  //! \name Convenient Key Comparison Functions Generated From key_less
  //! \{

  //! True if a < b ? "constructed" from key_less_()
  bool key_less(const key_type &a, const key_type &b) const { return key_less_(a, b); }

  //! True if a <= b ? constructed from key_less()
  bool key_lessequal(const key_type &a, const key_type &b) const { return !key_less_(b, a); }

  //! True if a > b ? constructed from key_less()
  bool key_greater(const key_type &a, const key_type &b) const { return key_less_(b, a); }

  //! True if a >= b ? constructed from key_less()
  bool key_greaterequal(const key_type &a, const key_type &b) const { return !key_less_(a, b); }

  //! True if a == b ? constructed from key_less(). This requires the <
  //! relation to be a total order, otherwise the B+ tree cannot be sorted.
  bool key_equal(const key_type &a, const key_type &b) const { return !key_less_(a, b) && !key_less_(b, a); }

  //! \}

 public:
  //! \name Allocators
  //! \{

  //! Return the base node allocator provided during construction.
  allocator_type get_allocator() const { return allocator_; }

  //! \}

 private:
  //! \name Node Object Allocation and Deallocation Functions
  //! \{

  //! Return an allocator for LeafNode objects.
  typename LeafNode::alloc_type leaf_node_allocator() { return typename LeafNode::alloc_type(allocator_); }

  //! Return an allocator for InnerNode objects.
  typename InnerNode::alloc_type inner_node_allocator() { return typename InnerNode::alloc_type(allocator_); }

  //! Allocate and initialize a leaf node
  LeafNode *allocate_leaf() {
    LeafNode *n = new (leaf_node_allocator().allocate(1)) LeafNode();
    n->initialize();
    stats_.leaves++;
    return n;
  }

  //! Allocate and initialize an inner node
  InnerNode *allocate_inner(unsigned short level) {
    InnerNode *n = new (inner_node_allocator().allocate(1)) InnerNode();
    n->initialize(level);
    stats_.inner_nodes++;
    return n;
  }

  //! Correctly free either inner or leaf node, destructs all contained key
  //! and value objects.
  void free_node(node *n) {
    if (n->is_leafnode()) {
      LeafNode *ln = static_cast<LeafNode *>(n);
      typename LeafNode::alloc_type a(leaf_node_allocator());
      std::allocator_traits<typename LeafNode::alloc_type>::destroy(a, ln);
      std::allocator_traits<typename LeafNode::alloc_type>::deallocate(a, ln, 1);
      stats_.leaves--;
    } else {
      InnerNode *in = static_cast<InnerNode *>(n);
      typename InnerNode::alloc_type a(inner_node_allocator());
      std::allocator_traits<typename InnerNode::alloc_type>::destroy(a, in);
      std::allocator_traits<typename InnerNode::alloc_type>::deallocate(a, in, 1);
      stats_.inner_nodes--;
    }
  }

  //! \}

 public:
  //! \name Fast Destruction of the B+ Tree
  //! \{

  //! Frees all key/data pairs and all nodes of the tree.
  void clear() {
    if (root_) {
      clear_recursive(root_);
      free_node(root_);

      root_ = nullptr;
      head_leaf_ = tail_leaf_ = nullptr;

      stats_ = tree_stats();
    }

    TLX_BTREE_ASSERT(stats_.size == 0);
  }

 private:
  //! Recursively free up nodes.
  void clear_recursive(node *n) {
    if (n->is_leafnode()) {
      LeafNode *leafnode = static_cast<LeafNode *>(n);

      for (unsigned short slot = 0; slot < leafnode->slotuse; ++slot) {
        // data objects are deleted by LeafNode's destructor
      }
    } else {
      InnerNode *innernode = static_cast<InnerNode *>(n);

      for (unsigned short slot = 0; slot < innernode->slotuse + 1; ++slot) {
        clear_recursive(innernode->childid[slot]);
        free_node(innernode->childid[slot]);
      }
    }
  }

  //! \}

 public:
  //! \name STL Iterator Construction Functions
  //! \{

  //! Constructs a read/data-write iterator that points to the first slot in
  //! the first leaf of the B+ tree.
  iterator begin() { return iterator(head_leaf_, 0); }

  //! Constructs a read/data-write iterator that points to the first invalid
  //! slot in the last leaf of the B+ tree.
  iterator end() { return iterator(tail_leaf_, tail_leaf_ ? tail_leaf_->slotuse : 0); }

  //! Constructs a read-only constant iterator that points to the first slot
  //! in the first leaf of the B+ tree.
  const_iterator begin() const { return const_iterator(head_leaf_, 0); }

  //! Constructs a read-only constant iterator that points to the first
  //! invalid slot in the last leaf of the B+ tree.
  const_iterator end() const { return const_iterator(tail_leaf_, tail_leaf_ ? tail_leaf_->slotuse : 0); }

  //! Constructs a read/data-write reverse iterator that points to the first
  //! invalid slot in the last leaf of the B+ tree. Uses STL magic.
  reverse_iterator rbegin() { return reverse_iterator(end()); }

  //! Constructs a read/data-write reverse iterator that points to the first
  //! slot in the first leaf of the B+ tree. Uses STL magic.
  reverse_iterator rend() { return reverse_iterator(begin()); }

  //! Constructs a read-only reverse iterator that points to the first
  //! invalid slot in the last leaf of the B+ tree. Uses STL magic.
  const_reverse_iterator rbegin() const { return const_reverse_iterator(end()); }

  //! Constructs a read-only reverse iterator that points to the first slot
  //! in the first leaf of the B+ tree. Uses STL magic.
  const_reverse_iterator rend() const { return const_reverse_iterator(begin()); }

  //! \}

 private:
  //! \name B+ Tree Node Binary Search Functions
  //! \{

  //! Searches for the first key in the node n greater or equal to key. Uses
  //! binary search with an optional linear self-verification. This is a
  //! template function, because the slotkey array is located at different
  //! places in LeafNode and InnerNode.
  template <typename node_type>
  unsigned short find_lower(const node_type *n, const key_type &key) const {
    if (sizeof(*n) > traits::binsearch_threshold) {
      if (n->slotuse == 0) return 0;

      unsigned short lo = 0, hi = n->slotuse;

      while (lo < hi) {
        unsigned short mid = (lo + hi) >> 1;

        if (key_lessequal(key, n->key(mid))) {
          hi = mid;  // key <= mid
        } else {
          lo = mid + 1;  // key > mid
        }
      }

      TLX_BTREE_PRINT("BTree::find_lower: on " << n << " key " << key << " -> " << lo << " / " << hi);

      // verify result using simple linear search
      if (self_verify) {
        unsigned short i = 0;
        while (i < n->slotuse && key_less(n->key(i), key)) ++i;

        TLX_BTREE_PRINT("BTree::find_lower: testfind: " << i);
        TLX_BTREE_ASSERT(i == lo);
      }

      return lo;
    } else  // for nodes <= binsearch_threshold do linear search.
    {
      unsigned short lo = 0;
      while (lo < n->slotuse && key_less(n->key(lo), key)) ++lo;
      return lo;
    }
  }

  //! Searches for the first key in the node n greater than key. Uses binary
  //! search with an optional linear self-verification. This is a template
  //! function, because the slotkey array is located at different places in
  //! LeafNode and InnerNode.
  template <typename node_type>
  unsigned short find_upper(const node_type *n, const key_type &key) const {
    if (sizeof(*n) > traits::binsearch_threshold) {
      if (n->slotuse == 0) return 0;

      unsigned short lo = 0, hi = n->slotuse;

      while (lo < hi) {
        unsigned short mid = (lo + hi) >> 1;

        if (key_less(key, n->key(mid))) {
          hi = mid;  // key < mid
        } else {
          lo = mid + 1;  // key >= mid
        }
      }

      TLX_BTREE_PRINT("BTree::find_upper: on " << n << " key " << key << " -> " << lo << " / " << hi);

      // verify result using simple linear search
      if (self_verify) {
        unsigned short i = 0;
        while (i < n->slotuse && key_lessequal(n->key(i), key)) ++i;

        TLX_BTREE_PRINT("BTree::find_upper testfind: " << i);
        TLX_BTREE_ASSERT(i == hi);
      }

      return lo;
    } else  // for nodes <= binsearch_threshold do linear search.
    {
      unsigned short lo = 0;
      while (lo < n->slotuse && key_lessequal(n->key(lo), key)) ++lo;
      return lo;
    }
  }

  //! \}

 public:
  //! \name Access Functions to the Item Count
  //! \{

  //! Return the number of key/data pairs in the B+ tree
  size_type size() const { return stats_.size; }

  //! Returns true if there is at least one key/data pair in the B+ tree
  bool empty() const { return (size() == size_type(0)); }

  //! Returns the largest possible size of the B+ Tree. This is just a
  //! function required by the STL standard, the B+ Tree can hold more items.
  size_type max_size() const { return size_type(-1); }

  //! Return a const reference to the current statistics.
  const struct tree_stats &get_stats() const { return stats_; }

  //! \}

 public:
  //! \name STL Access Functions Querying the Tree by Descending to a Leaf
  //! \{

  //! Non-STL function checking whether a key is in the B+ tree. The same as
  //! (find(k) != end()) or (count() != 0).
  bool exists(const key_type &key) const {
    const node *n = root_;
    if (!n) return false;

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_lower(inner, key);

      n = inner->childid[slot];
    }

    const LeafNode *leaf = static_cast<const LeafNode *>(n);

    unsigned short slot = find_lower(leaf, key);
    return (slot < leaf->slotuse && key_equal(key, leaf->key(slot)));
  }

  //! Tries to locate a key in the B+ tree and returns an iterator to the
  //! key/data slot if found. If unsuccessful it returns end().
  iterator find(const key_type &key) {
    node *n = root_;
    if (!n) return end();

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_lower(inner, key);

      n = inner->childid[slot];
    }

    LeafNode *leaf = static_cast<LeafNode *>(n);

    unsigned short slot = find_lower(leaf, key);
    return (slot < leaf->slotuse && key_equal(key, leaf->key(slot))) ? iterator(leaf, slot) : end();
  }

  //! Tries to locate a key in the B+ tree and returns an constant iterator to
  //! the key/data slot if found. If unsuccessful it returns end().
  const_iterator find(const key_type &key) const {
    const node *n = root_;
    if (!n) return end();

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_lower(inner, key);

      n = inner->childid[slot];
    }

    const LeafNode *leaf = static_cast<const LeafNode *>(n);

    unsigned short slot = find_lower(leaf, key);
    return (slot < leaf->slotuse && key_equal(key, leaf->key(slot))) ? const_iterator(leaf, slot) : end();
  }

  //! Tries to locate a key in the B+ tree and returns the number of identical
  //! key entries found.
  size_type count(const key_type &key) const {
    const node *n = root_;
    if (!n) return 0;

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_lower(inner, key);

      n = inner->childid[slot];
    }

    const LeafNode *leaf = static_cast<const LeafNode *>(n);

    unsigned short slot = find_lower(leaf, key);
    size_type num = 0;

    while (leaf && slot < leaf->slotuse && key_equal(key, leaf->key(slot))) {
      ++num;
      if (++slot >= leaf->slotuse) {
        leaf = leaf->next_leaf;
        slot = 0;
      }
    }

    return num;
  }

  //! Searches the B+ tree and returns an iterator to the first pair equal to
  //! or greater than key, or end() if all keys are smaller.
  iterator lower_bound(const key_type &key) {
    node *n = root_;
    if (!n) return end();

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_lower(inner, key);

      n = inner->childid[slot];
    }

    LeafNode *leaf = static_cast<LeafNode *>(n);

    unsigned short slot = find_lower(leaf, key);
    return iterator(leaf, slot);
  }

  //! Searches the B+ tree and returns a constant iterator to the first pair
  //! equal to or greater than key, or end() if all keys are smaller.
  const_iterator lower_bound(const key_type &key) const {
    const node *n = root_;
    if (!n) return end();

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_lower(inner, key);

      n = inner->childid[slot];
    }

    const LeafNode *leaf = static_cast<const LeafNode *>(n);

    unsigned short slot = find_lower(leaf, key);
    return const_iterator(leaf, slot);
  }

  //! Searches the B+ tree and returns an iterator to the first pair greater
  //! than key, or end() if all keys are smaller or equal.
  iterator upper_bound(const key_type &key) {
    node *n = root_;
    if (!n) return end();

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_upper(inner, key);

      n = inner->childid[slot];
    }

    LeafNode *leaf = static_cast<LeafNode *>(n);

    unsigned short slot = find_upper(leaf, key);
    return iterator(leaf, slot);
  }

  //! Searches the B+ tree and returns a constant iterator to the first pair
  //! greater than key, or end() if all keys are smaller or equal.
  const_iterator upper_bound(const key_type &key) const {
    const node *n = root_;
    if (!n) return end();

    while (!n->is_leafnode()) {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      unsigned short slot = find_upper(inner, key);

      n = inner->childid[slot];
    }

    const LeafNode *leaf = static_cast<const LeafNode *>(n);

    unsigned short slot = find_upper(leaf, key);
    return const_iterator(leaf, slot);
  }

  //! Searches the B+ tree and returns both lower_bound() and upper_bound().
  std::pair<iterator, iterator> equal_range(const key_type &key) {
    return std::pair<iterator, iterator>(lower_bound(key), upper_bound(key));
  }

  //! Searches the B+ tree and returns both lower_bound() and upper_bound().
  std::pair<const_iterator, const_iterator> equal_range(const key_type &key) const {
    return std::pair<const_iterator, const_iterator>(lower_bound(key), upper_bound(key));
  }

  //! \}

 public:
  //! \name B+ Tree Object Comparison Functions
  //! \{

  //! Equality relation of B+ trees of the same type. B+ trees of the same
  //! size and equal elements (both key and data) are considered equal. Beware
  //! of the random ordering of duplicate keys.
  bool operator==(const BTree &other) const {
    return (size() == other.size()) && std::equal(begin(), end(), other.begin());
  }

  //! Inequality relation. Based on operator==.
  bool operator!=(const BTree &other) const { return !(*this == other); }

  //! Total ordering relation of B+ trees of the same type. It uses
  //! std::lexicographical_compare() for the actual comparison of elements.
  bool operator<(const BTree &other) const {
    return std::lexicographical_compare(begin(), end(), other.begin(), other.end());
  }

  //! Greater relation. Based on operator<.
  bool operator>(const BTree &other) const { return other < *this; }

  //! Less-equal relation. Based on operator<.
  bool operator<=(const BTree &other) const { return !(other < *this); }

  //! Greater-equal relation. Based on operator<.
  bool operator>=(const BTree &other) const { return !(*this < other); }

  //! \}

 public:
  //! \name Fast Copy: Assign Operator and Copy Constructors
  //! \{

  //! Assignment operator. All the key/data pairs are copied.
  BTree &operator=(const BTree &other) {
    if (this != &other) {
      clear();

      key_less_ = other.key_comp();
      allocator_ = other.get_allocator();

      if (other.size() != 0) {
        stats_.leaves = stats_.inner_nodes = 0;
        if (other.root_) {
          root_ = copy_recursive(other.root_);
        }
        stats_ = other.stats_;
      }

      if (self_verify) verify();
    }
    return *this;
  }

  //! Copy constructor. The newly initialized B+ tree object will contain a
  //! copy of all key/data pairs.
  BTree(const BTree &other)
      : root_(nullptr),
        head_leaf_(nullptr),
        tail_leaf_(nullptr),
        stats_(other.stats_),
        key_less_(other.key_comp()),
        allocator_(other.get_allocator()) {
    if (size() > 0) {
      stats_.leaves = stats_.inner_nodes = 0;
      if (other.root_) {
        root_ = copy_recursive(other.root_);
      }
      if (self_verify) verify();
    }
  }

 private:
  //! Recursively copy nodes from another B+ tree object
  struct node *copy_recursive(const node *n) {
    if (n->is_leafnode()) {
      const LeafNode *leaf = static_cast<const LeafNode *>(n);
      LeafNode *newleaf = allocate_leaf();

      newleaf->slotuse = leaf->slotuse;
      std::copy(leaf->slotdata, leaf->slotdata + leaf->slotuse, newleaf->slotdata);

      if (head_leaf_ == nullptr) {
        head_leaf_ = tail_leaf_ = newleaf;
        newleaf->prev_leaf = newleaf->next_leaf = nullptr;
      } else {
        newleaf->prev_leaf = tail_leaf_;
        tail_leaf_->next_leaf = newleaf;
        tail_leaf_ = newleaf;
      }

      return newleaf;
    } else {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      InnerNode *newinner = allocate_inner(inner->level);

      newinner->slotuse = inner->slotuse;
      std::copy(inner->slotkey, inner->slotkey + inner->slotuse, newinner->slotkey);

      for (unsigned short slot = 0; slot <= inner->slotuse; ++slot) {
        newinner->childid[slot] = copy_recursive(inner->childid[slot]);
      }

      return newinner;
    }
  }

  //! \}

 public:
  //! \name Public Insertion Functions
  //! \{

  //! Attempt to insert a key/data pair into the B+ tree. If the tree does not
  //! allow duplicate keys, then the insert may fail if it is already present.
  std::pair<iterator, bool> insert(const value_type &x) { return insert_start(key_of_value::get(x), x); }

  //! Attempt to insert a key/data pair into the B+ tree. The iterator hint is
  //! currently ignored by the B+ tree insertion routine.
  iterator insert(iterator /* hint */, const value_type &x) { return insert_start(key_of_value::get(x), x).first; }

  //! Attempt to insert the range [first,last) of value_type pairs into the B+
  //! tree. Each key/data pair is inserted individually; to bulk load the
  //! tree, use a constructor with range.
  template <typename InputIterator>
  void insert(InputIterator first, InputIterator last) {
    InputIterator iter = first;
    while (iter != last) {
      insert(*iter);
      ++iter;
    }
  }

  //! \}

 private:
  //! \name Private Insertion Functions
  //! \{

  //! Start the insertion descent at the current root and handle root splits.
  //! Returns true if the item was inserted
  std::pair<iterator, bool> insert_start(const key_type &key, const value_type &value) {
    node *newchild = nullptr;
    key_type newkey = key_type();

    if (root_ == nullptr) {
      root_ = head_leaf_ = tail_leaf_ = allocate_leaf();
    }

    std::pair<iterator, bool> r = insert_descend(root_, key, value, &newkey, &newchild);

    if (newchild) {
      // this only occurs if insert_descend() could not insert the key
      // into the root node, this mean the root is full and a new root
      // needs to be created.
      InnerNode *newroot = allocate_inner(root_->level + 1);
      newroot->slotkey[0] = newkey;

      newroot->childid[0] = root_;
      newroot->childid[1] = newchild;

      newroot->slotuse = 1;

      root_ = newroot;
    }

    // increment size if the item was inserted
    if (r.second) ++stats_.size;

#ifdef TLX_BTREE_DEBUG
    if (debug) print(std::cout);
#endif

    if (self_verify) {
      verify();
      TLX_BTREE_ASSERT(exists(key));
    }

    return r;
  }

  /*!
   * Insert an item into the B+ tree.
   *
   * Descend down the nodes to a leaf, insert the key/data pair in a free
   * slot. If the node overflows, then it must be split and the new split node
   * inserted into the parent. Unroll / this splitting up to the root.
   */
  std::pair<iterator, bool> insert_descend(node *n, const key_type &key, const value_type &value, key_type *splitkey,
                                           node **splitnode) {
    if (!n->is_leafnode()) {
      InnerNode *inner = static_cast<InnerNode *>(n);

      key_type newkey = key_type();
      node *newchild = nullptr;

      unsigned short slot = find_lower(inner, key);

      TLX_BTREE_PRINT("BTree::insert_descend into " << inner->childid[slot]);

      std::pair<iterator, bool> r = insert_descend(inner->childid[slot], key, value, &newkey, &newchild);

      if (newchild) {
        TLX_BTREE_PRINT("BTree::insert_descend newchild"
                        << " with key " << newkey << " node " << newchild << " at slot " << slot);

        if (inner->is_full()) {
          split_inner_node(inner, splitkey, splitnode, slot);

          TLX_BTREE_PRINT("BTree::insert_descend done split_inner:"
                          << " putslot: " << slot << " putkey: " << newkey << " upkey: " << *splitkey);

#ifdef TLX_BTREE_DEBUG
          if (debug) {
            print_node(std::cout, inner);
            print_node(std::cout, *splitnode);
          }
#endif

          // check if insert slot is in the split sibling node
          TLX_BTREE_PRINT("BTree::insert_descend switch: " << slot << " > " << inner->slotuse + 1);

          if (slot == inner->slotuse + 1 && inner->slotuse < (*splitnode)->slotuse) {
            // special case when the insert slot matches the split
            // place between the two nodes, then the insert key
            // becomes the split key.

            TLX_BTREE_ASSERT(inner->slotuse + 1 < inner_slotmax);

            InnerNode *split = static_cast<InnerNode *>(*splitnode);

            // move the split key and it's datum into the left node
            inner->slotkey[inner->slotuse] = *splitkey;
            inner->childid[inner->slotuse + 1] = split->childid[0];
            inner->slotuse++;

            // set new split key and move corresponding datum into
            // right node
            split->childid[0] = newchild;
            *splitkey = newkey;

            return r;
          } else if (slot >= inner->slotuse + 1) {
            // in case the insert slot is in the newly create split
            // node, we reuse the code below.

            slot -= inner->slotuse + 1;
            inner = static_cast<InnerNode *>(*splitnode);
            TLX_BTREE_PRINT(
                "BTree::insert_descend switching to "
                "splitted node "
                << inner << " slot " << slot);
          }
        }

        // move items and put pointer to child node into correct slot
        TLX_BTREE_ASSERT(slot >= 0 && slot <= inner->slotuse);

        std::copy_backward(inner->slotkey + slot, inner->slotkey + inner->slotuse, inner->slotkey + inner->slotuse + 1);
        std::copy_backward(inner->childid + slot, inner->childid + inner->slotuse + 1,
                           inner->childid + inner->slotuse + 2);

        inner->slotkey[slot] = newkey;
        inner->childid[slot + 1] = newchild;
        inner->slotuse++;
      }

      return r;
    } else  // n->is_leafnode() == true
    {
      LeafNode *leaf = static_cast<LeafNode *>(n);

      unsigned short slot = find_lower(leaf, key);

      if (!allow_duplicates && slot < leaf->slotuse && key_equal(key, leaf->key(slot))) {
        return std::pair<iterator, bool>(iterator(leaf, slot), false);
      }

      if (leaf->is_full()) {
        split_leaf_node(leaf, splitkey, splitnode);

        // check if insert slot is in the split sibling node
        if (slot >= leaf->slotuse) {
          slot -= leaf->slotuse;
          leaf = static_cast<LeafNode *>(*splitnode);
        }
      }

      // move items and put data item into correct data slot
      TLX_BTREE_ASSERT(slot >= 0 && slot <= leaf->slotuse);

      std::copy_backward(leaf->slotdata + slot, leaf->slotdata + leaf->slotuse, leaf->slotdata + leaf->slotuse + 1);

      leaf->slotdata[slot] = value;
      leaf->slotuse++;

      if (splitnode && leaf != *splitnode && slot == leaf->slotuse - 1) {
        // special case: the node was split, and the insert is at the
        // last slot of the old node. then the splitkey must be updated.
        *splitkey = key;
      }

      return std::pair<iterator, bool>(iterator(leaf, slot), true);
    }
  }

  //! Split up a leaf node into two equally-filled sibling leaves. Returns the
  //! new nodes and it's insertion key in the two parameters.
  void split_leaf_node(LeafNode *leaf, key_type *out_newkey, node **out_newleaf) {
    TLX_BTREE_ASSERT(leaf->is_full());

    unsigned short mid = (leaf->slotuse >> 1);

    TLX_BTREE_PRINT("BTree::split_leaf_node on " << leaf);

    LeafNode *newleaf = allocate_leaf();

    newleaf->slotuse = leaf->slotuse - mid;

    newleaf->next_leaf = leaf->next_leaf;
    if (newleaf->next_leaf == nullptr) {
      TLX_BTREE_ASSERT(leaf == tail_leaf_);
      tail_leaf_ = newleaf;
    } else {
      newleaf->next_leaf->prev_leaf = newleaf;
    }

    std::copy(leaf->slotdata + mid, leaf->slotdata + leaf->slotuse, newleaf->slotdata);

    leaf->slotuse = mid;
    leaf->next_leaf = newleaf;
    newleaf->prev_leaf = leaf;

    *out_newkey = leaf->key(leaf->slotuse - 1);
    *out_newleaf = newleaf;
  }

  //! Split up an inner node into two equally-filled sibling nodes. Returns
  //! the new nodes and it's insertion key in the two parameters. Requires the
  //! slot of the item will be inserted, so the nodes will be the same size
  //! after the insert.
  void split_inner_node(InnerNode *inner, key_type *out_newkey, node **out_newinner, unsigned int addslot) {
    TLX_BTREE_ASSERT(inner->is_full());

    unsigned short mid = (inner->slotuse >> 1);

    TLX_BTREE_PRINT("BTree::split_inner: mid " << mid << " addslot " << addslot);

    // if the split is uneven and the overflowing item will be put into the
    // larger node, then the smaller split node may underflow
    if (addslot <= mid && mid > inner->slotuse - (mid + 1)) mid--;

    TLX_BTREE_PRINT("BTree::split_inner: mid " << mid << " addslot " << addslot);

    TLX_BTREE_PRINT("BTree::split_inner_node on " << inner << " into two nodes " << mid << " and "
                                                  << inner->slotuse - (mid + 1) << " sized");

    InnerNode *newinner = allocate_inner(inner->level);

    newinner->slotuse = inner->slotuse - (mid + 1);

    std::copy(inner->slotkey + mid + 1, inner->slotkey + inner->slotuse, newinner->slotkey);
    std::copy(inner->childid + mid + 1, inner->childid + inner->slotuse + 1, newinner->childid);

    inner->slotuse = mid;

    *out_newkey = inner->key(mid);
    *out_newinner = newinner;
  }

  //! \}

 public:
  //! \name Bulk Loader - Construct Tree from Sorted Sequence
  //! \{

  //! Bulk load a sorted range. Loads items into leaves and constructs a
  //! B-tree above them. The tree must be empty when calling this function.
  template <typename Iterator>
  void bulk_load(Iterator ibegin, Iterator iend) {
    TLX_BTREE_ASSERT(empty());

    stats_.size = iend - ibegin;

    // calculate number of leaves needed, round up.
    size_t num_items = iend - ibegin;
    size_t num_leaves = (num_items + leaf_slotmax - 1) / leaf_slotmax;

    TLX_BTREE_PRINT("BTree::bulk_load, level 0: "
                    << stats_.size << " items into " << num_leaves << " leaves with up to "
                    << ((iend - ibegin + num_leaves - 1) / num_leaves) << " items per leaf.");

    Iterator it = ibegin;
    for (size_t i = 0; i < num_leaves; ++i) {
      // allocate new leaf node
      LeafNode *leaf = allocate_leaf();

      // copy keys or (key,value) pairs into leaf nodes, uses template
      // switch leaf->set_slot().
      leaf->slotuse = static_cast<int>(num_items / (num_leaves - i));
      for (size_t s = 0; s < leaf->slotuse; ++s, ++it) leaf->set_slot(s, *it);

      if (tail_leaf_ != nullptr) {
        tail_leaf_->next_leaf = leaf;
        leaf->prev_leaf = tail_leaf_;
      } else {
        head_leaf_ = leaf;
      }
      tail_leaf_ = leaf;

      num_items -= leaf->slotuse;
    }

    TLX_BTREE_ASSERT(it == iend && num_items == 0);

    // if the btree is so small to fit into one leaf, then we're done.
    if (head_leaf_ == tail_leaf_) {
      root_ = head_leaf_;
      return;
    }

    TLX_BTREE_ASSERT(stats_.leaves == num_leaves);

    // create first level of inner nodes, pointing to the leaves.
    size_t num_parents = (num_leaves + (inner_slotmax + 1) - 1) / (inner_slotmax + 1);

    TLX_BTREE_PRINT("BTree::bulk_load, level 1: "
                    << num_leaves << " leaves in " << num_parents << " inner nodes with up to "
                    << ((num_leaves + num_parents - 1) / num_parents) << " leaves per inner node.");

    // save inner nodes and maxkey for next level.
    typedef std::pair<InnerNode *, const key_type *> nextlevel_type;
    nextlevel_type *nextlevel = new nextlevel_type[num_parents];

    LeafNode *leaf = head_leaf_;
    for (size_t i = 0; i < num_parents; ++i) {
      // allocate new inner node at level 1
      InnerNode *n = allocate_inner(1);

      n->slotuse = static_cast<int>(num_leaves / (num_parents - i));
      TLX_BTREE_ASSERT(n->slotuse > 0);
      // this counts keys, but an inner node has keys+1 children.
      --n->slotuse;

      // copy last key from each leaf and set child
      for (unsigned short s = 0; s < n->slotuse; ++s) {
        n->slotkey[s] = leaf->key(leaf->slotuse - 1);
        n->childid[s] = leaf;
        leaf = leaf->next_leaf;
      }
      n->childid[n->slotuse] = leaf;

      // track max key of any descendant.
      nextlevel[i].first = n;
      nextlevel[i].second = &leaf->key(leaf->slotuse - 1);

      leaf = leaf->next_leaf;
      num_leaves -= n->slotuse + 1;
    }

    TLX_BTREE_ASSERT(leaf == nullptr && num_leaves == 0);

    // recursively build inner nodes pointing to inner nodes.
    for (int level = 2; num_parents != 1; ++level) {
      size_t num_children = num_parents;
      num_parents = (num_children + (inner_slotmax + 1) - 1) / (inner_slotmax + 1);

      TLX_BTREE_PRINT("BTree::bulk_load, level "
                      << level << ": " << num_children << " children in " << num_parents << " inner nodes with up to "
                      << ((num_children + num_parents - 1) / num_parents) << " children per inner node.");

      size_t inner_index = 0;
      for (size_t i = 0; i < num_parents; ++i) {
        // allocate new inner node at level
        InnerNode *n = allocate_inner(level);

        n->slotuse = static_cast<int>(num_children / (num_parents - i));
        TLX_BTREE_ASSERT(n->slotuse > 0);
        // this counts keys, but an inner node has keys+1 children.
        --n->slotuse;

        // copy children and maxkeys from nextlevel
        for (unsigned short s = 0; s < n->slotuse; ++s) {
          n->slotkey[s] = *nextlevel[inner_index].second;
          n->childid[s] = nextlevel[inner_index].first;
          ++inner_index;
        }
        n->childid[n->slotuse] = nextlevel[inner_index].first;

        // reuse nextlevel array for parents, because we can overwrite
        // slots we've already consumed.
        nextlevel[i].first = n;
        nextlevel[i].second = nextlevel[inner_index].second;

        ++inner_index;
        num_children -= n->slotuse + 1;
      }

      TLX_BTREE_ASSERT(num_children == 0);
    }

    root_ = nextlevel[0].first;
    delete[] nextlevel;

    if (self_verify) verify();
  }

  //! \}

 private:
  //! \name Support Class Encapsulating Deletion Results
  //! \{

  //! Result flags of recursive deletion.
  enum result_flags_t {
    //! Deletion successful and no fix-ups necessary.
    btree_ok = 0,

    //! Deletion not successful because key was not found.
    btree_not_found = 1,

    //! Deletion successful, the last key was updated so parent slotkeys
    //! need updates.
    btree_update_lastkey = 2,

    //! Deletion successful, children nodes were merged and the parent needs
    //! to remove the empty node.
    btree_fixmerge = 4
  };

  //! B+ tree recursive deletion has much information which is needs to be
  //! passed upward.
  struct result_t {
    //! Merged result flags
    result_flags_t flags;

    //! The key to be updated at the parent's slot
    key_type lastkey;

    //! Constructor of a result with a specific flag, this can also be used
    //! as for implicit conversion.
    result_t(result_flags_t f = btree_ok)  // NOLINT
        : flags(f), lastkey() {}

    //! Constructor with a lastkey value.
    result_t(result_flags_t f, const key_type &k) : flags(f), lastkey(k) {}

    //! Test if this result object has a given flag set.
    bool has(result_flags_t f) const { return (flags & f) != 0; }

    //! Merge two results OR-ing the result flags and overwriting lastkeys.
    result_t &operator|=(const result_t &other) {
      flags = result_flags_t(flags | other.flags);

      // we overwrite existing lastkeys on purpose
      if (other.has(btree_update_lastkey)) lastkey = other.lastkey;

      return *this;
    }
  };

  //! \}

 public:
  //! \name Public Erase Functions
  //! \{

  //! Erases one (the first) of the key/data pairs associated with the given
  //! key.
  bool erase_one(const key_type &key) {
    TLX_BTREE_PRINT("BTree::erase_one(" << key << ") on btree size " << size());

    if (self_verify) verify();

    if (!root_) return false;

    result_t result = erase_one_descend(key, root_, nullptr, nullptr, nullptr, nullptr, nullptr, 0);

    if (!result.has(btree_not_found)) --stats_.size;

#ifdef TLX_BTREE_DEBUG
    if (debug) print(std::cout);
#endif
    if (self_verify) verify();

    return !result.has(btree_not_found);
  }

  //! Erases all the key/data pairs associated with the given key. This is
  //! implemented using erase_one().
  size_type erase(const key_type &key) {
    size_type c = 0;

    while (erase_one(key)) {
      ++c;
      if (!allow_duplicates) break;
    }

    return c;
  }

  //! Erase the key/data pair referenced by the iterator.
  void erase(iterator iter) {
    TLX_BTREE_PRINT("BTree::erase_iter(" << iter.curr_leaf << "," << iter.curr_slot << ") on btree size " << size());

    if (self_verify) verify();

    if (!root_) return;

    result_t result = erase_iter_descend(iter, root_, nullptr, nullptr, nullptr, nullptr, nullptr, 0);

    if (!result.has(btree_not_found)) --stats_.size;

#ifdef TLX_BTREE_DEBUG
    if (debug) print(std::cout);
#endif
    if (self_verify) verify();
  }

#ifdef BTREE_TODO
  //! Erase all key/data pairs in the range [first,last). This function is
  //! currently not implemented by the B+ Tree.
  void erase(iterator /* first */, iterator /* last */) { abort(); }
#endif

  //! \}

 private:
  //! \name Private Erase Functions
  //! \{

  /*!
   * Erase one (the first) key/data pair in the B+ tree matching key.
   *
   * Descends down the tree in search of key. During the descent the parent,
   * left and right siblings and their parents are computed and passed
   * down. Once the key/data pair is found, it is removed from the leaf. If
   * the leaf underflows 6 different cases are handled. These cases resolve
   * the underflow by shifting key/data pairs from adjacent sibling nodes,
   * merging two sibling nodes or trimming the tree.
   */
  result_t erase_one_descend(const key_type &key, node *curr, node *left, node *right, InnerNode *left_parent,
                             InnerNode *right_parent, InnerNode *parent, unsigned int parentslot) {
    if (curr->is_leafnode()) {
      LeafNode *leaf = static_cast<LeafNode *>(curr);
      LeafNode *left_leaf = static_cast<LeafNode *>(left);
      LeafNode *right_leaf = static_cast<LeafNode *>(right);

      unsigned short slot = find_lower(leaf, key);

      if (slot >= leaf->slotuse || !key_equal(key, leaf->key(slot))) {
        TLX_BTREE_PRINT("Could not find key " << key << " to erase.");

        return btree_not_found;
      }

      TLX_BTREE_PRINT("Found key in leaf " << curr << " at slot " << slot);

      std::copy(leaf->slotdata + slot + 1, leaf->slotdata + leaf->slotuse, leaf->slotdata + slot);

      leaf->slotuse--;

      result_t myres = btree_ok;

      // if the last key of the leaf was changed, the parent is notified
      // and updates the key of this leaf
      if (slot == leaf->slotuse) {
        if (parent && parentslot < parent->slotuse) {
          TLX_BTREE_ASSERT(parent->childid[parentslot] == curr);
          parent->slotkey[parentslot] = leaf->key(leaf->slotuse - 1);
        } else {
          if (leaf->slotuse >= 1) {
            TLX_BTREE_PRINT("Scheduling lastkeyupdate: key " << leaf->key(leaf->slotuse - 1));
            myres |= result_t(btree_update_lastkey, leaf->key(leaf->slotuse - 1));
          } else {
            TLX_BTREE_ASSERT(leaf == root_);
          }
        }
      }

      if (leaf->is_underflow() && !(leaf == root_ && leaf->slotuse >= 1)) {
        // determine what to do about the underflow

        // case : if this empty leaf is the root, then delete all nodes
        // and set root to nullptr.
        if (left_leaf == nullptr && right_leaf == nullptr) {
          TLX_BTREE_ASSERT(leaf == root_);
          TLX_BTREE_ASSERT(leaf->slotuse == 0);

          free_node(root_);

          root_ = leaf = nullptr;
          head_leaf_ = tail_leaf_ = nullptr;

          // will be decremented soon by insert_start()
          TLX_BTREE_ASSERT(stats_.size == 1);
          TLX_BTREE_ASSERT(stats_.leaves == 0);
          TLX_BTREE_ASSERT(stats_.inner_nodes == 0);

          return btree_ok;
        }
        // case : if both left and right leaves would underflow in case
        // of a shift, then merging is necessary. choose the more local
        // merger with our parent
        else if ((left_leaf == nullptr || left_leaf->is_few()) && (right_leaf == nullptr || right_leaf->is_few())) {
          if (left_parent == parent)
            myres |= merge_leaves(left_leaf, leaf, left_parent);
          else
            myres |= merge_leaves(leaf, right_leaf, right_parent);
        }
        // case : the right leaf has extra data, so balance right with
        // current
        else if ((left_leaf != nullptr && left_leaf->is_few()) && (right_leaf != nullptr && !right_leaf->is_few())) {
          if (right_parent == parent)
            myres |= shift_left_leaf(leaf, right_leaf, right_parent, parentslot);
          else
            myres |= merge_leaves(left_leaf, leaf, left_parent);
        }
        // case : the left leaf has extra data, so balance left with
        // current
        else if ((left_leaf != nullptr && !left_leaf->is_few()) && (right_leaf != nullptr && right_leaf->is_few())) {
          if (left_parent == parent)
            shift_right_leaf(left_leaf, leaf, left_parent, parentslot - 1);
          else
            myres |= merge_leaves(leaf, right_leaf, right_parent);
        }
        // case : both the leaf and right leaves have extra data and our
        // parent, choose the leaf with more data
        else if (left_parent == right_parent) {
          if (left_leaf->slotuse <= right_leaf->slotuse)
            myres |= shift_left_leaf(leaf, right_leaf, right_parent, parentslot);
          else
            shift_right_leaf(left_leaf, leaf, left_parent, parentslot - 1);
        } else {
          if (left_parent == parent)
            shift_right_leaf(left_leaf, leaf, left_parent, parentslot - 1);
          else
            myres |= shift_left_leaf(leaf, right_leaf, right_parent, parentslot);
        }
      }

      return myres;
    } else  // !curr->is_leafnode()
    {
      InnerNode *inner = static_cast<InnerNode *>(curr);
      InnerNode *left_inner = static_cast<InnerNode *>(left);
      InnerNode *right_inner = static_cast<InnerNode *>(right);

      node *myleft, *myright;
      InnerNode *myleft_parent, *myright_parent;

      unsigned short slot = find_lower(inner, key);

      if (slot == 0) {
        myleft = (left == nullptr) ? nullptr : static_cast<InnerNode *>(left)->childid[left->slotuse - 1];
        myleft_parent = left_parent;
      } else {
        myleft = inner->childid[slot - 1];
        myleft_parent = inner;
      }

      if (slot == inner->slotuse) {
        myright = (right == nullptr) ? nullptr : static_cast<InnerNode *>(right)->childid[0];
        myright_parent = right_parent;
      } else {
        myright = inner->childid[slot + 1];
        myright_parent = inner;
      }

      TLX_BTREE_PRINT("erase_one_descend into " << inner->childid[slot]);

      result_t result =
          erase_one_descend(key, inner->childid[slot], myleft, myright, myleft_parent, myright_parent, inner, slot);

      result_t myres = btree_ok;

      if (result.has(btree_not_found)) {
        return result;
      }

      if (result.has(btree_update_lastkey)) {
        if (parent && parentslot < parent->slotuse) {
          TLX_BTREE_PRINT("Fixing lastkeyupdate: key " << result.lastkey << " into parent " << parent
                                                       << " at parentslot " << parentslot);

          TLX_BTREE_ASSERT(parent->childid[parentslot] == curr);
          parent->slotkey[parentslot] = result.lastkey;
        } else {
          TLX_BTREE_PRINT("Forwarding lastkeyupdate: key " << result.lastkey);
          myres |= result_t(btree_update_lastkey, result.lastkey);
        }
      }

      if (result.has(btree_fixmerge)) {
        // either the current node or the next is empty and should be
        // removed
        if (inner->childid[slot]->slotuse != 0) slot++;

        // this is the child slot invalidated by the merge
        TLX_BTREE_ASSERT(inner->childid[slot]->slotuse == 0);

        free_node(inner->childid[slot]);

        std::copy(inner->slotkey + slot, inner->slotkey + inner->slotuse, inner->slotkey + slot - 1);
        std::copy(inner->childid + slot + 1, inner->childid + inner->slotuse + 1, inner->childid + slot);

        inner->slotuse--;

        if (inner->level == 1) {
          // fix split key for children leaves
          slot--;
          LeafNode *child = static_cast<LeafNode *>(inner->childid[slot]);
          inner->slotkey[slot] = child->key(child->slotuse - 1);
        }
      }

      if (inner->is_underflow() && !(inner == root_ && inner->slotuse >= 1)) {
        // case: the inner node is the root and has just one child. that
        // child becomes the new root
        if (left_inner == nullptr && right_inner == nullptr) {
          TLX_BTREE_ASSERT(inner == root_);
          TLX_BTREE_ASSERT(inner->slotuse == 0);

          root_ = inner->childid[0];

          inner->slotuse = 0;
          free_node(inner);

          return btree_ok;
        }
        // case : if both left and right leaves would underflow in case
        // of a shift, then merging is necessary. choose the more local
        // merger with our parent
        else if ((left_inner == nullptr || left_inner->is_few()) && (right_inner == nullptr || right_inner->is_few())) {
          if (left_parent == parent)
            myres |= merge_inner(left_inner, inner, left_parent, parentslot - 1);
          else
            myres |= merge_inner(inner, right_inner, right_parent, parentslot);
        }
        // case : the right leaf has extra data, so balance right with
        // current
        else if ((left_inner != nullptr && left_inner->is_few()) &&
                 (right_inner != nullptr && !right_inner->is_few())) {
          if (right_parent == parent)
            shift_left_inner(inner, right_inner, right_parent, parentslot);
          else
            myres |= merge_inner(left_inner, inner, left_parent, parentslot - 1);
        }
        // case : the left leaf has extra data, so balance left with
        // current
        else if ((left_inner != nullptr && !left_inner->is_few()) &&
                 (right_inner != nullptr && right_inner->is_few())) {
          if (left_parent == parent)
            shift_right_inner(left_inner, inner, left_parent, parentslot - 1);
          else
            myres |= merge_inner(inner, right_inner, right_parent, parentslot);
        }
        // case : both the leaf and right leaves have extra data and our
        // parent, choose the leaf with more data
        else if (left_parent == right_parent) {
          if (left_inner->slotuse <= right_inner->slotuse)
            shift_left_inner(inner, right_inner, right_parent, parentslot);
          else
            shift_right_inner(left_inner, inner, left_parent, parentslot - 1);
        } else {
          if (left_parent == parent)
            shift_right_inner(left_inner, inner, left_parent, parentslot - 1);
          else
            shift_left_inner(inner, right_inner, right_parent, parentslot);
        }
      }

      return myres;
    }
  }

  /*!
   * Erase one key/data pair referenced by an iterator in the B+ tree.
   *
   * Descends down the tree in search of an iterator. During the descent the
   * parent, left and right siblings and their parents are computed and passed
   * down. The difficulty is that the iterator contains only a pointer to a
   * LeafNode, which means that this function must do a recursive depth first
   * search for that leaf node in the subtree containing all pairs of the same
   * key. This subtree can be very large, even the whole tree, though in
   * practice it would not make sense to have so many duplicate keys.
   *
   * Once the referenced key/data pair is found, it is removed from the leaf
   * and the same underflow cases are handled as in erase_one_descend.
   */
  result_t erase_iter_descend(const iterator &iter, node *curr, node *left, node *right, InnerNode *left_parent,
                              InnerNode *right_parent, InnerNode *parent, unsigned int parentslot) {
    if (curr->is_leafnode()) {
      LeafNode *leaf = static_cast<LeafNode *>(curr);
      LeafNode *left_leaf = static_cast<LeafNode *>(left);
      LeafNode *right_leaf = static_cast<LeafNode *>(right);

      // if this is not the correct leaf, get next step in recursive
      // search
      if (leaf != iter.curr_leaf) {
        return btree_not_found;
      }

      if (iter.curr_slot >= leaf->slotuse) {
        TLX_BTREE_PRINT("Could not find iterator (" << iter.curr_leaf << "," << iter.curr_slot
                                                    << ") to erase. Invalid leaf node?");

        return btree_not_found;
      }

      unsigned short slot = iter.curr_slot;

      TLX_BTREE_PRINT("Found iterator in leaf " << curr << " at slot " << slot);

      std::copy(leaf->slotdata + slot + 1, leaf->slotdata + leaf->slotuse, leaf->slotdata + slot);

      leaf->slotuse--;

      result_t myres = btree_ok;

      // if the last key of the leaf was changed, the parent is notified
      // and updates the key of this leaf
      if (slot == leaf->slotuse) {
        if (parent && parentslot < parent->slotuse) {
          TLX_BTREE_ASSERT(parent->childid[parentslot] == curr);
          parent->slotkey[parentslot] = leaf->key(leaf->slotuse - 1);
        } else {
          if (leaf->slotuse >= 1) {
            TLX_BTREE_PRINT("Scheduling lastkeyupdate: key " << leaf->key(leaf->slotuse - 1));
            myres |= result_t(btree_update_lastkey, leaf->key(leaf->slotuse - 1));
          } else {
            TLX_BTREE_ASSERT(leaf == root_);
          }
        }
      }

      if (leaf->is_underflow() && !(leaf == root_ && leaf->slotuse >= 1)) {
        // determine what to do about the underflow

        // case : if this empty leaf is the root, then delete all nodes
        // and set root to nullptr.
        if (left_leaf == nullptr && right_leaf == nullptr) {
          TLX_BTREE_ASSERT(leaf == root_);
          TLX_BTREE_ASSERT(leaf->slotuse == 0);

          free_node(root_);

          root_ = leaf = nullptr;
          head_leaf_ = tail_leaf_ = nullptr;

          // will be decremented soon by insert_start()
          TLX_BTREE_ASSERT(stats_.size == 1);
          TLX_BTREE_ASSERT(stats_.leaves == 0);
          TLX_BTREE_ASSERT(stats_.inner_nodes == 0);

          return btree_ok;
        }
        // case : if both left and right leaves would underflow in case
        // of a shift, then merging is necessary. choose the more local
        // merger with our parent
        else if ((left_leaf == nullptr || left_leaf->is_few()) && (right_leaf == nullptr || right_leaf->is_few())) {
          if (left_parent == parent)
            myres |= merge_leaves(left_leaf, leaf, left_parent);
          else
            myres |= merge_leaves(leaf, right_leaf, right_parent);
        }
        // case : the right leaf has extra data, so balance right with
        // current
        else if ((left_leaf != nullptr && left_leaf->is_few()) && (right_leaf != nullptr && !right_leaf->is_few())) {
          if (right_parent == parent) {
            myres |= shift_left_leaf(leaf, right_leaf, right_parent, parentslot);
          } else {
            myres |= merge_leaves(left_leaf, leaf, left_parent);
          }
        }
        // case : the left leaf has extra data, so balance left with
        // current
        else if ((left_leaf != nullptr && !left_leaf->is_few()) && (right_leaf != nullptr && right_leaf->is_few())) {
          if (left_parent == parent) {
            shift_right_leaf(left_leaf, leaf, left_parent, parentslot - 1);
          } else {
            myres |= merge_leaves(leaf, right_leaf, right_parent);
          }
        }
        // case : both the leaf and right leaves have extra data and our
        // parent, choose the leaf with more data
        else if (left_parent == right_parent) {
          if (left_leaf->slotuse <= right_leaf->slotuse) {
            myres |= shift_left_leaf(leaf, right_leaf, right_parent, parentslot);
          } else {
            shift_right_leaf(left_leaf, leaf, left_parent, parentslot - 1);
          }
        } else {
          if (left_parent == parent) {
            shift_right_leaf(left_leaf, leaf, left_parent, parentslot - 1);
          } else {
            myres |= shift_left_leaf(leaf, right_leaf, right_parent, parentslot);
          }
        }
      }

      return myres;
    } else  // !curr->is_leafnode()
    {
      InnerNode *inner = static_cast<InnerNode *>(curr);
      InnerNode *left_inner = static_cast<InnerNode *>(left);
      InnerNode *right_inner = static_cast<InnerNode *>(right);

      // find first slot below which the searched iterator might be
      // located.

      result_t result;
      unsigned short slot = find_lower(inner, iter.key());

      while (slot <= inner->slotuse) {
        node *myleft, *myright;
        InnerNode *myleft_parent, *myright_parent;

        if (slot == 0) {
          myleft = (left == nullptr) ? nullptr : static_cast<InnerNode *>(left)->childid[left->slotuse - 1];
          myleft_parent = left_parent;
        } else {
          myleft = inner->childid[slot - 1];
          myleft_parent = inner;
        }

        if (slot == inner->slotuse) {
          myright = (right == nullptr) ? nullptr : static_cast<InnerNode *>(right)->childid[0];
          myright_parent = right_parent;
        } else {
          myright = inner->childid[slot + 1];
          myright_parent = inner;
        }

        TLX_BTREE_PRINT("erase_iter_descend into " << inner->childid[slot]);

        result =
            erase_iter_descend(iter, inner->childid[slot], myleft, myright, myleft_parent, myright_parent, inner, slot);

        if (!result.has(btree_not_found)) break;

        // continue recursive search for leaf on next slot

        if (slot < inner->slotuse && key_less(inner->slotkey[slot], iter.key())) return btree_not_found;

        ++slot;
      }

      if (slot > inner->slotuse) return btree_not_found;

      result_t myres = btree_ok;

      if (result.has(btree_update_lastkey)) {
        if (parent && parentslot < parent->slotuse) {
          TLX_BTREE_PRINT("Fixing lastkeyupdate: key " << result.lastkey << " into parent " << parent
                                                       << " at parentslot " << parentslot);

          TLX_BTREE_ASSERT(parent->childid[parentslot] == curr);
          parent->slotkey[parentslot] = result.lastkey;
        } else {
          TLX_BTREE_PRINT("Forwarding lastkeyupdate: key " << result.lastkey);
          myres |= result_t(btree_update_lastkey, result.lastkey);
        }
      }

      if (result.has(btree_fixmerge)) {
        // either the current node or the next is empty and should be
        // removed
        if (inner->childid[slot]->slotuse != 0) slot++;

        // this is the child slot invalidated by the merge
        TLX_BTREE_ASSERT(inner->childid[slot]->slotuse == 0);

        free_node(inner->childid[slot]);

        std::copy(inner->slotkey + slot, inner->slotkey + inner->slotuse, inner->slotkey + slot - 1);
        std::copy(inner->childid + slot + 1, inner->childid + inner->slotuse + 1, inner->childid + slot);

        inner->slotuse--;

        if (inner->level == 1) {
          // fix split key for children leaves
          slot--;
          LeafNode *child = static_cast<LeafNode *>(inner->childid[slot]);
          inner->slotkey[slot] = child->key(child->slotuse - 1);
        }
      }

      if (inner->is_underflow() && !(inner == root_ && inner->slotuse >= 1)) {
        // case: the inner node is the root and has just one
        // child. that child becomes the new root
        if (left_inner == nullptr && right_inner == nullptr) {
          TLX_BTREE_ASSERT(inner == root_);
          TLX_BTREE_ASSERT(inner->slotuse == 0);

          root_ = inner->childid[0];

          inner->slotuse = 0;
          free_node(inner);

          return btree_ok;
        }
        // case : if both left and right leaves would underflow in case
        // of a shift, then merging is necessary. choose the more local
        // merger with our parent
        else if ((left_inner == nullptr || left_inner->is_few()) && (right_inner == nullptr || right_inner->is_few())) {
          if (left_parent == parent) {
            myres |= merge_inner(left_inner, inner, left_parent, parentslot - 1);
          } else {
            myres |= merge_inner(inner, right_inner, right_parent, parentslot);
          }
        }
        // case : the right leaf has extra data, so balance right with
        // current
        else if ((left_inner != nullptr && left_inner->is_few()) &&
                 (right_inner != nullptr && !right_inner->is_few())) {
          if (right_parent == parent) {
            shift_left_inner(inner, right_inner, right_parent, parentslot);
          } else {
            myres |= merge_inner(left_inner, inner, left_parent, parentslot - 1);
          }
        }
        // case : the left leaf has extra data, so balance left with
        // current
        else if ((left_inner != nullptr && !left_inner->is_few()) &&
                 (right_inner != nullptr && right_inner->is_few())) {
          if (left_parent == parent) {
            shift_right_inner(left_inner, inner, left_parent, parentslot - 1);
          } else {
            myres |= merge_inner(inner, right_inner, right_parent, parentslot);
          }
        }
        // case : both the leaf and right leaves have extra data and our
        // parent, choose the leaf with more data
        else if (left_parent == right_parent) {
          if (left_inner->slotuse <= right_inner->slotuse) {
            shift_left_inner(inner, right_inner, right_parent, parentslot);
          } else {
            shift_right_inner(left_inner, inner, left_parent, parentslot - 1);
          }
        } else {
          if (left_parent == parent) {
            shift_right_inner(left_inner, inner, left_parent, parentslot - 1);
          } else {
            shift_left_inner(inner, right_inner, right_parent, parentslot);
          }
        }
      }

      return myres;
    }
  }

  //! Merge two leaf nodes. The function moves all key/data pairs from right
  //! to left and sets right's slotuse to zero. The right slot is then removed
  //! by the calling parent node.
  result_t merge_leaves(LeafNode *left, LeafNode *right, InnerNode *parent) {
    TLX_BTREE_PRINT("Merge leaf nodes " << left << " and " << right << " with common parent " << parent << ".");
    (void)parent;

    TLX_BTREE_ASSERT(left->is_leafnode() && right->is_leafnode());
    TLX_BTREE_ASSERT(parent->level == 1);

    TLX_BTREE_ASSERT(left->slotuse + right->slotuse < leaf_slotmax);

    std::copy(right->slotdata, right->slotdata + right->slotuse, left->slotdata + left->slotuse);

    left->slotuse += right->slotuse;

    left->next_leaf = right->next_leaf;
    if (left->next_leaf)
      left->next_leaf->prev_leaf = left;
    else
      tail_leaf_ = left;

    right->slotuse = 0;

    return btree_fixmerge;
  }

  //! Merge two inner nodes. The function moves all key/childid pairs from
  //! right to left and sets right's slotuse to zero. The right slot is then
  //! removed by the calling parent node.
  static result_t merge_inner(InnerNode *left, InnerNode *right, InnerNode *parent, unsigned int parentslot) {
    TLX_BTREE_PRINT("Merge inner nodes " << left << " and " << right << " with common parent " << parent << ".");

    TLX_BTREE_ASSERT(left->level == right->level);
    TLX_BTREE_ASSERT(parent->level == left->level + 1);

    TLX_BTREE_ASSERT(parent->childid[parentslot] == left);

    TLX_BTREE_ASSERT(left->slotuse + right->slotuse < inner_slotmax);

    if (self_verify) {
      // find the left node's slot in the parent's children
      unsigned int leftslot = 0;
      while (leftslot <= parent->slotuse && parent->childid[leftslot] != left) ++leftslot;

      TLX_BTREE_ASSERT(leftslot < parent->slotuse);
      TLX_BTREE_ASSERT(parent->childid[leftslot] == left);
      TLX_BTREE_ASSERT(parent->childid[leftslot + 1] == right);

      TLX_BTREE_ASSERT(parentslot == leftslot);
    }

    // retrieve the decision key from parent
    left->slotkey[left->slotuse] = parent->slotkey[parentslot];
    left->slotuse++;

    // copy over keys and children from right
    std::copy(right->slotkey, right->slotkey + right->slotuse, left->slotkey + left->slotuse);
    std::copy(right->childid, right->childid + right->slotuse + 1, left->childid + left->slotuse);

    left->slotuse += right->slotuse;
    right->slotuse = 0;

    return btree_fixmerge;
  }

  //! Balance two leaf nodes. The function moves key/data pairs from right to
  //! left so that both nodes are equally filled. The parent node is updated
  //! if possible.
  static result_t shift_left_leaf(LeafNode *left, LeafNode *right, InnerNode *parent, unsigned int parentslot) {
    TLX_BTREE_ASSERT(left->is_leafnode() && right->is_leafnode());
    TLX_BTREE_ASSERT(parent->level == 1);

    TLX_BTREE_ASSERT(left->next_leaf == right);
    TLX_BTREE_ASSERT(left == right->prev_leaf);

    TLX_BTREE_ASSERT(left->slotuse < right->slotuse);
    TLX_BTREE_ASSERT(parent->childid[parentslot] == left);

    unsigned int shiftnum = (right->slotuse - left->slotuse) >> 1;

    TLX_BTREE_PRINT("Shifting (leaf) " << shiftnum << " entries to left " << left << " from right " << right
                                       << " with common parent " << parent << ".");

    TLX_BTREE_ASSERT(left->slotuse + shiftnum < leaf_slotmax);

    // copy the first items from the right node to the last slot in the left
    // node.

    std::copy(right->slotdata, right->slotdata + shiftnum, left->slotdata + left->slotuse);

    left->slotuse += shiftnum;

    // shift all slots in the right node to the left

    std::copy(right->slotdata + shiftnum, right->slotdata + right->slotuse, right->slotdata);

    right->slotuse -= shiftnum;

    // fixup parent
    if (parentslot < parent->slotuse) {
      parent->slotkey[parentslot] = left->key(left->slotuse - 1);
      return btree_ok;
    } else {  // the update is further up the tree
      return result_t(btree_update_lastkey, left->key(left->slotuse - 1));
    }
  }

  //! Balance two inner nodes. The function moves key/data pairs from right to
  //! left so that both nodes are equally filled. The parent node is updated
  //! if possible.
  static void shift_left_inner(InnerNode *left, InnerNode *right, InnerNode *parent, unsigned int parentslot) {
    TLX_BTREE_ASSERT(left->level == right->level);
    TLX_BTREE_ASSERT(parent->level == left->level + 1);

    TLX_BTREE_ASSERT(left->slotuse < right->slotuse);
    TLX_BTREE_ASSERT(parent->childid[parentslot] == left);

    unsigned int shiftnum = (right->slotuse - left->slotuse) >> 1;

    TLX_BTREE_PRINT("Shifting (inner) " << shiftnum << " entries to left " << left << " from right " << right
                                        << " with common parent " << parent << ".");

    TLX_BTREE_ASSERT(left->slotuse + shiftnum < inner_slotmax);

    if (self_verify) {
      // find the left node's slot in the parent's children and compare to
      // parentslot

      unsigned int leftslot = 0;
      while (leftslot <= parent->slotuse && parent->childid[leftslot] != left) ++leftslot;

      TLX_BTREE_ASSERT(leftslot < parent->slotuse);
      TLX_BTREE_ASSERT(parent->childid[leftslot] == left);
      TLX_BTREE_ASSERT(parent->childid[leftslot + 1] == right);

      TLX_BTREE_ASSERT(leftslot == parentslot);
    }

    // copy the parent's decision slotkey and childid to the first new key
    // on the left
    left->slotkey[left->slotuse] = parent->slotkey[parentslot];
    left->slotuse++;

    // copy the other items from the right node to the last slots in the
    // left node.
    std::copy(right->slotkey, right->slotkey + shiftnum - 1, left->slotkey + left->slotuse);
    std::copy(right->childid, right->childid + shiftnum, left->childid + left->slotuse);

    left->slotuse += shiftnum - 1;

    // fixup parent
    parent->slotkey[parentslot] = right->slotkey[shiftnum - 1];

    // shift all slots in the right node
    std::copy(right->slotkey + shiftnum, right->slotkey + right->slotuse, right->slotkey);
    std::copy(right->childid + shiftnum, right->childid + right->slotuse + 1, right->childid);

    right->slotuse -= shiftnum;
  }

  //! Balance two leaf nodes. The function moves key/data pairs from left to
  //! right so that both nodes are equally filled. The parent node is updated
  //! if possible.
  static void shift_right_leaf(LeafNode *left, LeafNode *right, InnerNode *parent, unsigned int parentslot) {
    TLX_BTREE_ASSERT(left->is_leafnode() && right->is_leafnode());
    TLX_BTREE_ASSERT(parent->level == 1);

    TLX_BTREE_ASSERT(left->next_leaf == right);
    TLX_BTREE_ASSERT(left == right->prev_leaf);
    TLX_BTREE_ASSERT(parent->childid[parentslot] == left);

    TLX_BTREE_ASSERT(left->slotuse > right->slotuse);

    unsigned int shiftnum = (left->slotuse - right->slotuse) >> 1;

    TLX_BTREE_PRINT("Shifting (leaf) " << shiftnum << " entries to right " << right << " from left " << left
                                       << " with common parent " << parent << ".");

    if (self_verify) {
      // find the left node's slot in the parent's children
      unsigned int leftslot = 0;
      while (leftslot <= parent->slotuse && parent->childid[leftslot] != left) ++leftslot;

      TLX_BTREE_ASSERT(leftslot < parent->slotuse);
      TLX_BTREE_ASSERT(parent->childid[leftslot] == left);
      TLX_BTREE_ASSERT(parent->childid[leftslot + 1] == right);

      TLX_BTREE_ASSERT(leftslot == parentslot);
    }

    // shift all slots in the right node

    TLX_BTREE_ASSERT(right->slotuse + shiftnum < leaf_slotmax);

    std::copy_backward(right->slotdata, right->slotdata + right->slotuse, right->slotdata + right->slotuse + shiftnum);

    right->slotuse += shiftnum;

    // copy the last items from the left node to the first slot in the right
    // node.
    std::copy(left->slotdata + left->slotuse - shiftnum, left->slotdata + left->slotuse, right->slotdata);

    left->slotuse -= shiftnum;

    parent->slotkey[parentslot] = left->key(left->slotuse - 1);
  }

  //! Balance two inner nodes. The function moves key/data pairs from left to
  //! right so that both nodes are equally filled. The parent node is updated
  //! if possible.
  static void shift_right_inner(InnerNode *left, InnerNode *right, InnerNode *parent, unsigned int parentslot) {
    TLX_BTREE_ASSERT(left->level == right->level);
    TLX_BTREE_ASSERT(parent->level == left->level + 1);

    TLX_BTREE_ASSERT(left->slotuse > right->slotuse);
    TLX_BTREE_ASSERT(parent->childid[parentslot] == left);

    unsigned int shiftnum = (left->slotuse - right->slotuse) >> 1;

    TLX_BTREE_PRINT("Shifting (leaf) " << shiftnum << " entries to right " << right << " from left " << left
                                       << " with common parent " << parent << ".");

    if (self_verify) {
      // find the left node's slot in the parent's children
      unsigned int leftslot = 0;
      while (leftslot <= parent->slotuse && parent->childid[leftslot] != left) ++leftslot;

      TLX_BTREE_ASSERT(leftslot < parent->slotuse);
      TLX_BTREE_ASSERT(parent->childid[leftslot] == left);
      TLX_BTREE_ASSERT(parent->childid[leftslot + 1] == right);

      TLX_BTREE_ASSERT(leftslot == parentslot);
    }

    // shift all slots in the right node

    TLX_BTREE_ASSERT(right->slotuse + shiftnum < inner_slotmax);

    std::copy_backward(right->slotkey, right->slotkey + right->slotuse, right->slotkey + right->slotuse + shiftnum);
    std::copy_backward(right->childid, right->childid + right->slotuse + 1,
                       right->childid + right->slotuse + 1 + shiftnum);

    right->slotuse += shiftnum;

    // copy the parent's decision slotkey and childid to the last new key on
    // the right
    right->slotkey[shiftnum - 1] = parent->slotkey[parentslot];

    // copy the remaining last items from the left node to the first slot in
    // the right node.
    std::copy(left->slotkey + left->slotuse - shiftnum + 1, left->slotkey + left->slotuse, right->slotkey);
    std::copy(left->childid + left->slotuse - shiftnum + 1, left->childid + left->slotuse + 1, right->childid);

    // copy the first to-be-removed key from the left node to the parent's
    // decision slot
    parent->slotkey[parentslot] = left->slotkey[left->slotuse - shiftnum];

    left->slotuse -= shiftnum;
  }

  //! \}

#ifdef TLX_BTREE_DEBUG

 public:
  //! \name Debug Printing
  //! \{

  //! Print out the B+ tree structure with keys onto the given ostream. This
  //! function requires that the header is compiled with TLX_BTREE_DEBUG and
  //! that key_type is printable via std::ostream.
  void print(std::ostream &os) const {
    if (root_) {
      print_node(os, root_, 0, true);
    }
  }

  //! Print out only the leaves via the double linked list.
  void print_leaves(std::ostream &os) const {
    os << "leaves:" << std::endl;

    const LeafNode *n = head_leaf_;

    while (n) {
      os << "  " << n << std::endl;

      n = n->next_leaf;
    }
  }

 private:
  //! Recursively descend down the tree and print out nodes.
  static void print_node(std::ostream &os, const node *node, unsigned int depth = 0, bool recursive = false) {
    for (unsigned int i = 0; i < depth; i++) os << "  ";

    os << "node " << node << " level " << node->level << " slotuse " << node->slotuse << std::endl;

    if (node->is_leafnode()) {
      const LeafNode *leafnode = static_cast<const LeafNode *>(node);

      for (unsigned int i = 0; i < depth; i++) os << "  ";
      os << "  leaf prev " << leafnode->prev_leaf << " next " << leafnode->next_leaf << std::endl;

      for (unsigned int i = 0; i < depth; i++) os << "  ";

      for (unsigned short slot = 0; slot < leafnode->slotuse; ++slot) {
        // os << leafnode->key(slot) << " "
        //    << "(data: " << leafnode->slotdata[slot] << ") ";
        os << leafnode->key(slot) << "  ";
      }
      os << std::endl;
    } else {
      const InnerNode *innernode = static_cast<const InnerNode *>(node);

      for (unsigned int i = 0; i < depth; i++) os << "  ";

      for (unsigned short slot = 0; slot < innernode->slotuse; ++slot) {
        os << "(" << innernode->childid[slot] << ") " << innernode->slotkey[slot] << " ";
      }
      os << "(" << innernode->childid[innernode->slotuse] << ")" << std::endl;

      if (recursive) {
        for (unsigned short slot = 0; slot < innernode->slotuse + 1; ++slot) {
          print_node(os, innernode->childid[slot], depth + 1, recursive);
        }
      }
    }
  }

  //! \}
#endif

 public:
  //! \name Verification of B+ Tree Invariants
  //! \{

  //! Run a thorough verification of all B+ tree invariants. The program
  //! aborts via tlx_die_unless() if something is wrong.
  void verify() const {
    key_type minkey, maxkey;
    tree_stats vstats;

    if (root_) {
      verify_node(root_, &minkey, &maxkey, vstats);

      tlx_die_unless(vstats.size == stats_.size);
      tlx_die_unless(vstats.leaves == stats_.leaves);
      tlx_die_unless(vstats.inner_nodes == stats_.inner_nodes);

      verify_leaflinks();
    }
  }

 private:
  //! Recursively descend down the tree and verify each node
  void verify_node(const node *n, key_type *minkey, key_type *maxkey, tree_stats &vstats) const {
    TLX_BTREE_PRINT("verifynode " << n);

    if (n->is_leafnode()) {
      const LeafNode *leaf = static_cast<const LeafNode *>(n);

      tlx_die_unless(leaf == root_ || !leaf->is_underflow());
      tlx_die_unless(leaf->slotuse > 0);

      for (unsigned short slot = 0; slot < leaf->slotuse - 1; ++slot) {
        tlx_die_unless(key_lessequal(leaf->key(slot), leaf->key(slot + 1)));
      }

      *minkey = leaf->key(0);
      *maxkey = leaf->key(leaf->slotuse - 1);

      vstats.leaves++;
      vstats.size += leaf->slotuse;
    } else  // !n->is_leafnode()
    {
      const InnerNode *inner = static_cast<const InnerNode *>(n);
      vstats.inner_nodes++;

      tlx_die_unless(inner == root_ || !inner->is_underflow());
      tlx_die_unless(inner->slotuse > 0);

      for (unsigned short slot = 0; slot < inner->slotuse - 1; ++slot) {
        tlx_die_unless(key_lessequal(inner->key(slot), inner->key(slot + 1)));
      }

      for (unsigned short slot = 0; slot <= inner->slotuse; ++slot) {
        const node *subnode = inner->childid[slot];
        key_type subminkey = key_type();
        key_type submaxkey = key_type();

        tlx_die_unless(subnode->level + 1 == inner->level);
        verify_node(subnode, &subminkey, &submaxkey, vstats);

        TLX_BTREE_PRINT("verify subnode " << subnode << ": " << subminkey << " - " << submaxkey);

        if (slot == 0)
          *minkey = subminkey;
        else
          tlx_die_unless(key_greaterequal(subminkey, inner->key(slot - 1)));

        if (slot == inner->slotuse)
          *maxkey = submaxkey;
        else
          tlx_die_unless(key_equal(inner->key(slot), submaxkey));

        if (inner->level == 1 && slot < inner->slotuse) {
          // children are leaves and must be linked together in the
          // correct order
          const LeafNode *leafa = static_cast<const LeafNode *>(inner->childid[slot]);
          const LeafNode *leafb = static_cast<const LeafNode *>(inner->childid[slot + 1]);

          tlx_die_unless(leafa->next_leaf == leafb);
          tlx_die_unless(leafa == leafb->prev_leaf);
        }
        if (inner->level == 2 && slot < inner->slotuse) {
          // verify leaf links between the adjacent inner nodes
          const InnerNode *parenta = static_cast<const InnerNode *>(inner->childid[slot]);
          const InnerNode *parentb = static_cast<const InnerNode *>(inner->childid[slot + 1]);

          const LeafNode *leafa = static_cast<const LeafNode *>(parenta->childid[parenta->slotuse]);
          const LeafNode *leafb = static_cast<const LeafNode *>(parentb->childid[0]);

          tlx_die_unless(leafa->next_leaf == leafb);
          tlx_die_unless(leafa == leafb->prev_leaf);
        }
      }
    }
  }

  //! Verify the double linked list of leaves.
  void verify_leaflinks() const {
    const LeafNode *n = head_leaf_;

    tlx_die_unless(n->level == 0);
    tlx_die_unless(!n || n->prev_leaf == nullptr);

    unsigned int testcount = 0;

    while (n) {
      tlx_die_unless(n->level == 0);
      tlx_die_unless(n->slotuse > 0);

      for (unsigned short slot = 0; slot < n->slotuse - 1; ++slot) {
        tlx_die_unless(key_lessequal(n->key(slot), n->key(slot + 1)));
      }

      testcount += n->slotuse;

      if (n->next_leaf) {
        tlx_die_unless(key_lessequal(n->key(n->slotuse - 1), n->next_leaf->key(0)));

        tlx_die_unless(n == n->next_leaf->prev_leaf);
      } else {
        tlx_die_unless(tail_leaf_ == n);
      }

      n = n->next_leaf;
    }

    tlx_die_unless(testcount == size());
  }

  //! \}
};

//! \}
//! \}

}  // namespace tlx

#endif  // !TLX_CONTAINER_BTREE_HEADER

/******************************************************************************/