AIOJ

An AVL tree is a balanced binary search tree. Because many algorithm textbooks introduce AVL trees at excessive length, many people get the impression that AVL trees are complicated and impractical. In fact, the principles of AVL trees are simple, and their implementation is not complicated.

Properties

An empty binary tree is an AVL tree
If T is an AVL tree, then its left and right subtrees are also AVL trees, and $|h(ls) - h(rs)| \leq 1$ , where h is the height of its left or right subtree
The tree height is $O(\log n)$

Balance factor: height of the right subtree minus height of the left subtree

Proof of the height bound

Let $f_n$ be the minimum number of nodes in an AVL tree of height $n$ . Then

f_n= \begin{cases} 1&(n=1)\\ 2&(n=2)\\ f_{n-1}+f_{n-2}+1& (n>2) \end{cases}

By solving the nonhomogeneous linear recurrence with constant coefficients, $\{f_n+1\}$ is a Fibonacci sequence. The closed form of $f_n$ is:

f_n=\frac{5+2\sqrt{5}}{5}\left(\frac{1+\sqrt{5}}{2}\right)^n+\frac{5-2\sqrt{5}}{5}\left(\frac{1-\sqrt{5}}{2}\right)^n-1

The Fibonacci sequence grows exponentially, and for tree height $n$ we have:

n<\log_{\frac{1+\sqrt{5}}{2}} (f_n+1)<\frac{3}{2}\log_2 (f_n+1)

Therefore, the height of an AVL tree is $O(\log f_n)$ , where $f_n$ is the number of nodes.

Process

Inserting a Node

As with a BST (binary search tree), first perform an unsuccessful search to determine the insertion position. After inserting the node, use the balance factor to decide whether rebalancing is needed.

Deleting a Node

Deletion is similar to deletion in a BST: swap the node with its successor and then delete it.

Deletion changes the tree height and balance factors, so the change must be adjusted along the path from the deleted node to the root.

Maintaining Balance

Inserting or deleting a node may violate Property 2 of an AVL tree. Therefore, the tree must be maintained along the path from the inserted/deleted node to the root. If Property 2 no longer holds for some node, the height can have changed by at most 1 because only one node was inserted or deleted, so the absolute value of that node's balance factor is at most 2. By symmetry, we discuss only the case where the left subtree is 2 higher than the right subtree, namely $h(B)-h(E)=2$ in the figure below. We then distinguish two cases according to the relationship between $h(A)$ and $h(C)$ . Note that because balance is maintained from bottom to top, Property 2 still holds for every descendant of node D.

Case 1: The Height of A Is at Least the Height of C

Let $h(E)=x$ . Then

\begin{cases} h(B)=x+2\\ h(A)=x+1\\ x\leq h(C)\leq x+1 \end{cases}

Here, $h(C)\geq x$ follows because node B satisfies Property 2, so the difference between $h(C)$ and $h(A)$ is at most 1. We now perform a right rotation on node D (the rotation is the same as for other types of balanced binary search trees), as shown below.

Clearly, the heights of nodes A, C, and E do not change, and

\begin{cases} 0\leq h(C)-h(E)\leq 1\\ x+1\leq h'(D)=\max(h(C),h(E))+1=h(C)+1\leq x+2\\ 0\leq h'(D)-h(A)\leq 1 \end{cases}

Therefore, nodes B and D also satisfy Property 2 after the rotation.

Case 2: The Height of A Is Less Than the Height of C

Let $h(E)=x$ . Similarly, we have

\begin{cases} h(B)=x+2\\ h(C)=x+1\\ h(A)=x \end{cases}

We first perform a left rotation on node B and then a right rotation on node D, as shown below.

Clearly, the heights of nodes A and E do not change. The new right child of B and the new left child of D are respectively the original left and right children of C, so

\begin{cases} x-1\leq h'(rs_B),h'(ls_D)\leq x\\ 0\leq h(A)-h'(rs_B)\leq 1\\ 0\leq h(E)-h'(ls_D)\leq 1\\ h'(B)=\max(h(A),h'(rs_B))+1=x+1\\ h'(D)=\max(h(E),h'(ls_D))+1=x+1\\ h'(B)-h'(D)=0 \end{cases}

Therefore, nodes B, C, and D also satisfy Property 2 after the rotation.

Balance Maintenance: Pseudocode

\begin{array}{ll} 1 & \textbf{function } \mathrm{MaintainBalance}(p) \\ 2 & \qquad l \gets ls_p, r \gets rs_p \\ 3 & \qquad \textbf{if } h(l)-h(r)=2 \\ 4 & \qquad\qquad \textbf{if } h(ls_l) \ge h(rs_l) \\ 5 & \qquad\qquad\qquad \mathrm{RightRotate}(p) \\ 6 & \qquad\qquad \textbf{else} \\ 7 & \qquad\qquad\qquad \mathrm{LeftRotate}(l) \\ 8 & \qquad\qquad\qquad \mathrm{RightRotate}(p) \\ 9 & \qquad \textbf{else if } h(l)-h(r)=-2 \\ 10 & \qquad\qquad \textbf{if } h(ls_r) \le h(rs_r) \\ 11 & \qquad\qquad\qquad \mathrm{LeftRotate}(p) \\ 12 & \qquad\qquad \textbf{else} \\ 13 & \qquad\qquad\qquad \mathrm{RightRotate}(r) \\ 14 & \qquad\qquad\qquad \mathrm{LeftRotate}(p) \\ \end{array}

As with other balanced binary search trees, information such as node heights and subtree sizes in an AVL tree must be maintained during rotations.

Other Operations

Other operations on AVL trees (Predecessor, Successor, Select, Rank, and so on) are the same as those on ordinary binary search trees.

Reference Code

The following code implements a Map with an AVL tree, that is, an ordered map without remapping:

Reference Code

/**
 * @brief An AVLTree-based map implementation
 * @details The map is sorted according to the natural ordering of its
 *  keys or by a {@code Compare} function provided; This implementation
 *  provides guaranteed log(n) time cost for the contains, get, insert
 *  and remove operations.
 */

#ifndef AVLTREE_MAP_HPP
#define AVLTREE_MAP_HPP

#include <cassert>
#include <cstddef>
#include <cstdint>
#include <functional>
#include <memory>
#include <stack>
#include <utility>
#include <vector>

/**
 * An AVLTree-based map implementation
 * https://en.wikipedia.org/wiki/AVL_tree
 * @tparam Key the type of keys maintained by this map
 * @tparam Value the type of mapped values
 * @tparam Compare
 */
template <typename Key, typename Value, typename Compare = std::less<Key> >
class AvlTreeMap {
 private:
  using USize = size_t;
  using Factor = int64_t;

  Compare compare = Compare();

 public:
  struct Entry {
    Key key;
    Value value;

    bool operator==(const Entry &rhs) const noexcept {
      return this->key == rhs.key && this->value == rhs.value;
    }

    bool operator!=(const Entry &rhs) const noexcept {
      return this->key != rhs.key || this->value != rhs.value;
    }
  };

 private:
  struct Node {
    using Ptr = std::shared_ptr<Node>;
    using Provider = const std::function<Ptr(void)> &;
    using Consumer = const std::function<void(const Ptr &)> &;

    Key key;
    Value value{};

    Ptr left = nullptr;
    Ptr right = nullptr;

    USize height = 1;

    explicit Node(Key k) : key(std::move(k)) {}

    explicit Node(Key k, Value v) : key(std::move(k)), value(std::move(v)) {}

    ~Node() = default;

    inline bool isLeaf() const noexcept {
      return this->left == nullptr && this->right == nullptr;
    }

    inline void updateHeight() noexcept {
      if (this->isLeaf()) {
        this->height = 1;
      } else if (this->left == nullptr) {
        this->height = this->right->height + 1;
      } else if (this->right == nullptr) {
        this->height = this->left->height + 1;
      } else {
        this->height = std::max(left->height, right->height) + 1;
      }
    }

    inline Factor factor() const noexcept {
      if (this->isLeaf()) {
        return 0;
      } else if (this->left == nullptr) {
        return (Factor)this->right->height;
      } else if (this->right == nullptr) {
        return (Factor) - this->left->height;
      } else {
        return (Factor)(this->right->height - this->left->height);
      }
    }

    inline Entry entry() const { return Entry{key, value}; }

    static Ptr from(const Key &k) { return std::make_shared<Node>(Node(k)); }

    static Ptr from(const Key &k, const Value &v) {
      return std::make_shared<Node>(Node(k, v));
    }
  };

  using NodePtr = typename Node::Ptr;
  using ConstNodePtr = const NodePtr &;
  using NodeProvider = typename Node::Provider;
  using NodeConsumer = typename Node::Consumer;

  NodePtr root = nullptr;
  USize count = 0;

  using K = const Key &;
  using V = const Value &;

 public:
  using EntryList = std::vector<Entry>;
  using KeyValueConsumer = const std::function<void(K, V)> &;
  using MutKeyValueConsumer = const std::function<void(K, Value &)> &;
  using KeyValueFilter = const std::function<bool(K, V)> &;

  class NoSuchMappingException : protected std::exception {
   private:
    const char *message;

   public:
    explicit NoSuchMappingException(const char *msg) : message(msg) {}

    const char *what() const noexcept override { return message; }
  };

  AvlTreeMap() noexcept = default;

  /**
   * Returns the number of entries in this map.
   * @return size_t
   */
  inline USize size() const noexcept { return this->count; }

  /**
   * Returns true if this collection contains no elements.
   * @return bool
   */
  inline bool empty() const noexcept { return this->count == 0; }

  /**
   * Removes all of the elements from this map.
   */
  void clear() noexcept {
    this->root = nullptr;
    this->count = 0;
  }

  /**
   * Returns the value to which the specified key is mapped; If this map
   * contains no mapping for the key, a {@code NoSuchMappingException} will
   * be thrown.
   * @param key
   * @return AvlTreeMap<Key, Value>::Value
   * @throws NoSuchMappingException
   */
  Value get(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("Invalid key");
    } else {
      NodePtr node = this->getNode(this->root, key);
      if (node != nullptr) {
        return node->value;
      } else {
        throw NoSuchMappingException("Invalid key");
      }
    }
  }

  /**
   * Returns the value to which the specified key is mapped; If this map
   * contains no mapping for the key, a new mapping with a default value
   * will be inserted.
   * @param key
   * @return AvlTreeMap<Key, Value>::Value &
   */
  Value &getOrDefault(K key) {
    if (this->root == nullptr) {
      this->root = Node::from(key);
      this->count += 1;
      return this->root->value;
    } else {
      return this
          ->getNodeOrProvide(this->root, key,
                             [&key]() { return Node::from(key); })
          ->value;
    }
  }

  /**
   * Returns true if this map contains a mapping for the specified key.
   * @param key
   * @return bool
   */
  bool contains(K key) const {
    return this->getNode(this->root, key) != nullptr;
  }

  /**
   * Associates the specified value with the specified key in this map.
   * @param key
   * @param value
   */
  void insert(K key, V value) {
    if (this->root == nullptr) {
      this->root = Node::from(key, value);
      this->count += 1;
    } else {
      this->insert(this->root, key, value);
    }
  }

  /**
   * If the specified key is not already associated with a value, associates
   * it with the given value and returns true, else returns false.
   * @param key
   * @param value
   * @return bool
   */
  bool insertIfAbsent(K key, V value) {
    USize sizeBeforeInsertion = this->size();
    if (this->root == nullptr) {
      this->root = Node::from(key, value);
      this->count += 1;
    } else {
      this->insert(this->root, key, value, false);
    }
    return this->size() > sizeBeforeInsertion;
  }

  /**
   * If the specified key is not already associated with a value, associates
   * it with the given value and returns the value, else returns the associated
   * value.
   * @param key
   * @param value
   * @return
   */
  Value &getOrInsert(K key, V value) {
    if (this->root == nullptr) {
      this->root = Node::from(key, value);
      this->count += 1;
      return root->value;
    } else {
      NodePtr node = getNodeOrProvide(this->root, key,
                                      [&]() { return Node::from(key, value); });
      return node->value;
    }
  }

  Value operator[](K key) const { return this->get(key); }

  Value &operator[](K key) { return this->getOrDefault(key); }

  /**
   * Removes the mapping for a key from this map if it is present;
   * Returns true if the mapping is present else returns false
   * @param key the key of the mapping
   * @return bool
   */
  bool remove(K key) {
    if (this->root == nullptr) {
      return false;
    } else {
      return this->remove(this->root, key, [](ConstNodePtr) {});
    }
  }

  /**
   * Removes the mapping for a key from this map if it is present and returns
   * the value which is mapped to the key; If this map contains no mapping for
   * the key, a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return AvlTreeMap<Key, Value>::Value
   * @throws NoSuchMappingException
   */
  Value getAndRemove(K key) {
    Value result;
    NodeConsumer action = [&](ConstNodePtr node) { result = node->value; };

    if (root == nullptr) {
      throw NoSuchMappingException("Invalid key");
    } else {
      if (remove(this->root, key, action)) {
        return result;
      } else {
        throw NoSuchMappingException("Invalid key");
      }
    }
  }

  /**
   * Gets the entry corresponding to the specified key; if no such entry
   * exists, returns the entry for the least key greater than the specified
   * key; if no such entry exists (i.e., the greatest key in the Tree is less
   * than the specified key), a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return AvlTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getCeilingEntry(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No ceiling entry in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (key == node->key) {
        return node->entry();
      }

      if (compare(key, node->key)) {
        /* key < node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          return node->entry();
        }
      } else {
        /* key > node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No ceiling entry in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->right) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException("No ceiling entry in this map");
            }
          }

          return parent->entry();
        }
      }
    }

    throw NoSuchMappingException("No ceiling entry in this map");
  }

  /**
   * Gets the entry corresponding to the specified key; if no such entry exists,
   * returns the entry for the greatest key less than the specified key;
   * if no such entry exists, a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return AvlTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getFloorEntry(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No floor entry exists in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (key == node->key) {
        return node->entry();
      }

      if (compare(key, node->key)) {
        /* key < node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No floor entry exists in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->left) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException("No floor entry exists in this map");
            }
          }

          return parent->entry();
        }
      } else {
        /* key > node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          return node->entry();
        }
      }
    }

    throw NoSuchMappingException("No floor entry exists in this map");
  }

  /**
   * Gets the entry for the least key greater than the specified
   * key; if no such entry exists, returns the entry for the least
   * key greater than the specified key; if no such entry exists,
   * a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return AvlTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getHigherEntry(K key) {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No higher entry exists in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (compare(key, node->key)) {
        /* key < node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          return node->entry();
        }
      } else {
        /* key >= node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No higher entry exists in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->right) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException(
                  "No higher entry exists in this map");
            }
          }

          return parent->entry();
        }
      }
    }

    throw NoSuchMappingException("No higher entry exists in this map");
  }

  /**
   * Returns the entry for the greatest key less than the specified key; if
   * no such entry exists (i.e., the least key in the Tree is greater than
   * the specified key), a {@code NoSuchMappingException} will be thrown.
   * @param key
   * @return AvlTreeMap<Key, Value>::Entry
   * @throws NoSuchMappingException
   */
  Entry getLowerEntry(K key) const {
    if (this->root == nullptr) {
      throw NoSuchMappingException("No lower entry exists in this map");
    }

    NodePtr node = this->root;
    std::stack<NodePtr> ancestors;

    while (node != nullptr) {
      if (compare(key, node->key) || key == node->key) {
        /* key <= node->key */
        if (node->left != nullptr) {
          ancestors.push(node);
          node = node->left;
        } else {
          if (ancestors.empty()) {
            throw NoSuchMappingException("No lower entry exists in this map");
          }

          NodePtr parent = ancestors.top();
          ancestors.pop();

          while (node == parent->left) {
            node = parent;
            if (!ancestors.empty()) {
              parent = ancestors.top();
              ancestors.pop();
            } else {
              throw NoSuchMappingException("No lower entry exists in this map");
            }
          }

          return parent->entry();
        }
      } else {
        /* key > node->key */
        if (node->right != nullptr) {
          ancestors.push(node);
          node = node->right;
        } else {
          return node->entry();
        }
      }
    }

    throw NoSuchMappingException("No lower entry exists in this map");
  }

  /**
   * Remove all entries that satisfy the filter condition.
   * @param filter
   */
  void removeAll(KeyValueFilter filter) {
    std::vector<Key> keys;
    this->inorderTraversal([&](ConstNodePtr node) {
      if (filter(node->key, node->value)) {
        keys.push_back(node->key);
      }
    });
    for (const Key &key : keys) {
      this->remove(key);
    }
  }

  /**
   * Performs the given action for each key and value entry in this map.
   * The value is immutable for the action.
   * @param action
   */
  void forEach(KeyValueConsumer action) const {
    this->inorderTraversal(
        [&](ConstNodePtr node) { action(node->key, node->value); });
  }

  /**
   * Performs the given action for each key and value entry in this map.
   * The value is mutable for the action.
   * @param action
   */
  void forEachMut(MutKeyValueConsumer action) {
    this->inorderTraversal(
        [&](ConstNodePtr node) { action(node->key, node->value); });
  }

  /**
   * Returns a list containing all of the entries in this map.
   * @return AvlTreeMap<Key, Value>::EntryList
   */
  EntryList toEntryList() const {
    EntryList entryList;
    this->inorderTraversal(
        [&](ConstNodePtr node) { entryList.push_back(node->entry()); });
    return entryList;
  }

 private:
  static NodePtr rotateLeft(ConstNodePtr node) {
    // clang-format off
    //     |                       |
    //     N                       S
    //    / \     l-rotate(N)     / \
    //   L   S    ==========>    N   R
    //      / \                 / \
    //     M   R               L   M
    NodePtr successor = node->right;
    // clang-format on
    node->right = successor->left;
    successor->left = node;

    node->updateHeight();
    successor->updateHeight();

    return successor;
  }

  static NodePtr rotateRight(ConstNodePtr node) {
    // clang-format off
    //       |                   |
    //       N                   S
    //      / \   r-rotate(N)   / \
    //     S   R  ==========>  L   N
    //    / \                     / \
    //   L   M                   M   R
    NodePtr successor = node->left;
    // clang-format on
    node->left = successor->right;
    successor->right = node;

    node->updateHeight();
    successor->updateHeight();

    return successor;
  }

  static void swapNode(NodePtr &lhs, NodePtr &rhs) {
    std::swap(lhs->key, rhs->key);
    std::swap(lhs->value, rhs->value);
    std::swap(lhs, rhs);
  }

  static void fixBalance(NodePtr &node) {
    if (node->factor() < -1) {
      if (node->left->factor() < 0) {
        // clang-format off
        //  Left-Left Case
        //       |
        //       C                 |
        //      /   r-rotate(C)    B
        //     B    ==========>   / \
        //    /                  A   C
        //   A
        // clang-format on
        node = rotateRight(node);
      } else {
        // clang-format off
        //  Left-Right Case
        //     |                   |
        //     C                   C                 |
        //    /   l-rotate(A)     /   r-rotate(C)    B
        //   A    ==========>    B    ==========>   / \
        //    \                 /                  A   C
        //     B               A
        // clang-format on
        node->left = rotateLeft(node->left);
        node = rotateRight(node);
      }
    } else if (node->factor() > 1) {
      if (node->right->factor() > 0) {
        // clang-format off
        //  Right-Right Case
        //   |
        //   C                     |
        //    \     l-rotate(C)    B
        //     B    ==========>   / \
        //      \                A   C
        //       A
        // clang-format on
        node = rotateLeft(node);
      } else {
        // clang-format off
        //  Right-Left Case
        //   |                 |
        //   A                 A                     |
        //    \   r-rotate(C)   \     l-rotate(A)    B
        //     C  ==========>    B    ==========>   / \
        //    /                   \                A   C
        //   B                     C
        // clang-format on
        node->right = rotateRight(node->right);
        node = rotateLeft(node);
      }
    }
  }

  NodePtr getNodeOrProvide(NodePtr &node, K key, NodeProvider provide) {
    assert(node != nullptr);

    if (key == node->key) {
      return node;
    }

    assert(key != node->key);

    NodePtr result;

    if (compare(key, node->key)) {
      /* key < node->key */
      if (node->left == nullptr) {
        result = node->left = provide();
        this->count += 1;
        node->updateHeight();
      } else {
        result = getNodeOrProvide(node->left, key, provide);
        node->updateHeight();
        fixBalance(node);
      }
    } else {
      /* key > node->key */
      if (node->right == nullptr) {
        result = node->right = provide();
        this->count += 1;
        node->updateHeight();
      } else {
        result = getNodeOrProvide(node->right, key, provide);
        node->updateHeight();
        fixBalance(node);
      }
    }

    return result;
  }

  NodePtr getNode(ConstNodePtr node, K key) const {
    assert(node != nullptr);

    if (key == node->key) {
      return node;
    }

    if (compare(key, node->key)) {
      /* key < node->key */
      return node->left == nullptr ? nullptr : getNode(node->left, key);
    } else {
      /* key > node->key */
      return node->right == nullptr ? nullptr : getNode(node->right, key);
    }
  }

  void insert(NodePtr &node, K key, V value, bool replace = true) {
    assert(node != nullptr);

    if (key == node->key) {
      if (replace) {
        node->value = value;
      }
      return;
    }

    assert(key != node->key);

    if (compare(key, node->key)) {
      /* key < node->key */
      if (node->left == nullptr) {
        node->left = Node::from(key, value);
        this->count += 1;
        node->updateHeight();
      } else {
        insert(node->left, key, value, replace);
        node->updateHeight();
        fixBalance(node);
      }
    } else {
      /* key > node->key */
      if (node->right == nullptr) {
        node->right = Node::from(key, value);
        this->count += 1;
        node->updateHeight();
      } else {
        insert(node->right, key, value, replace);
        node->updateHeight();
        fixBalance(node);
      }
    }
  }

  bool remove(NodePtr &node, K key, NodeConsumer action) {
    assert(node != nullptr);

    if (key != node->key) {
      if (compare(key, node->key)) {
        /* key < node->key */
        NodePtr &left = node->left;
        if (left != nullptr && remove(left, key, action)) {
          node->updateHeight();
          fixBalance(node);
          return true;
        } else {
          return false;
        }
      } else {
        /* key > node->key */
        NodePtr &right = node->right;
        if (right != nullptr && remove(right, key, action)) {
          node->updateHeight();
          fixBalance(node);
          return true;
        } else {
          return false;
        }
      }
    }

    assert(key == node->key);
    action(node);

    if (node->isLeaf()) {
      // Case 1: no child
      node = nullptr;
    } else if (node->right == nullptr) {
      // clang-format off
      // Case 2: left child only
      //     P
      //     |  remove(N)  P
      //     N  ========>  |
      //    /              L
      //   L
      // clang-format on
      node = node->left;
      node->updateHeight();
    } else if (node->left == nullptr) {
      // clang-format off
      // Case 3: right child only
      //   P
      //   |    remove(N)  P
      //   N    ========>  |
      //    \              R
      //     R
      // clang-format on
      node = node->right;
      node->updateHeight();
    } else if (node->right->left == nullptr) {
      // clang-format off
      // Case 4: both left and right child, right child has no left child
      //    |                 |
      //    N    remove(N)    R
      //   / \   ========>   /
      //  L   R             L
      // clang-format on
      NodePtr right = node->right;
      swapNode(node, right);
      right->right = node->right;
      node = right;
      node->updateHeight();
      fixBalance(node);
    } else {
      // clang-format off
      // Case 5: both left and right child, right child is not a leaf
      //   Step 1. find the node N with the smallest key
      //           and its parent P on the right subtree
      //   Step 2. swap S and N
      //   Step 3. remove node N like Case 1 or Case 3
      //   Step 4. update height for P
      //     |                  |
      //     N                  S                 |
      //    / \                / \                S
      //   L  ..  swap(N, S)  L  ..  remove(N)   / \
      //       |  =========>      |  ========>  L  ..
      //       P                  P                 |
      //      / \                / \                P
      //     S  ..              N  ..              / \
      //      \                  \                R  ..
      //       R                  R
      // clang-format on

      // Step 1
      NodePtr successor = node->right;
      NodePtr parent = node;
      while (successor->left != nullptr) {
        parent = successor;
        successor = parent->left;
      }
      // Step 2
      swapNode(node, successor);
      // Step 3
      parent->left = node->right;
      // Restore node
      node = successor;
      // Step 4
      parent->updateHeight();
    }

    this->count -= 1;
    return true;
  }

  void inorderTraversal(NodeConsumer action) const {
    if (this->root == nullptr) {
      return;
    }

    std::stack<NodePtr> stack;
    NodePtr node = this->root;

    while (node != nullptr || !stack.empty()) {
      while (node != nullptr) {
        stack.push(node);
        node = node->left;
      }
      if (!stack.empty()) {
        node = stack.top();
        stack.pop();
        action(node);
        node = node->right;
      }
    }
  }
};

#endif  // AVLTREE_MAP_HPP

AVL tree

Properties

Process

Inserting a Node

Deleting a Node

Maintaining Balance

Case 1: The Height of A Is at Least the Height of C

Case 2: The Height of A Is Less Than the Height of C

Other Operations

Reference Code

Further Reading