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7.5   AVL tree *

In the "Binary Search Tree" section, we mentioned that after multiple insertions and removals, a binary search tree might degrade to a linked list. In such cases, the time complexity of all operations degrades from \(O(\log n)\) to \(O(n)\).

As shown in the Figure 7-24 , after two node removal operations, this binary search tree will degrade into a linked list.

Degradation of an AVL tree after removing nodes

Figure 7-24   Degradation of an AVL tree after removing nodes

For example, in the perfect binary tree shown in the Figure 7-25 , after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.

Degradation of an AVL tree after inserting nodes

Figure 7-25   Degradation of an AVL tree after inserting nodes

In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the "AVL Tree" in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at \(O(\log n)\) level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.

7.5.1   Common terminology in AVL trees

An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a "balanced binary search tree".

1.   Node height

Since the operations related to AVL trees require obtaining node heights, we need to add a height variable to the node class:

class TreeNode:
    """AVL tree node"""
    def __init__(self, val: int):
        self.val: int = val                 # Node value
        self.height: int = 0                # Node height
        self.left: TreeNode | None = None   # Left child reference
        self.right: TreeNode | None = None  # Right child reference

/* AVL tree node */
struct TreeNode {
    int val{};          // Node value
    int height = 0;     // Node height
    TreeNode *left{};   // Left child
    TreeNode *right{};  // Right child
    TreeNode() = default;
    explicit TreeNode(int x) : val(x){}
};

/* AVL tree node */
class TreeNode {
    public int val;        // Node value
    public int height;     // Node height
    public TreeNode left;  // Left child
    public TreeNode right; // Right child
    public TreeNode(int x) { val = x; }
}

/* AVL tree node */
class TreeNode(int? x) {
    public int? val = x;    // Node value
    public int height;      // Node height
    public TreeNode? left;  // Left child reference
    public TreeNode? right; // Right child reference
}

/* AVL tree node */
type TreeNode struct {
    Val    int       // Node value
    Height int       // Node height
    Left   *TreeNode // Left child reference
    Right  *TreeNode // Right child reference
}

/* AVL tree node */
class TreeNode {
    var val: Int // Node value
    var height: Int // Node height
    var left: TreeNode? // Left child
    var right: TreeNode? // Right child

    init(x: Int) {
        val = x
        height = 0
    }
}

/* AVL tree node */
class TreeNode {
    val; // Node value
    height; // Node height
    left; // Left child pointer
    right; // Right child pointer
    constructor(val, left, right, height) {
        this.val = val === undefined ? 0 : val;
        this.height = height === undefined ? 0 : height;
        this.left = left === undefined ? null : left;
        this.right = right === undefined ? null : right;
    }
}

/* AVL tree node */
class TreeNode {
    val: number;            // Node value
    height: number;         // Node height
    left: TreeNode | null;  // Left child pointer
    right: TreeNode | null; // Right child pointer
    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {
        this.val = val === undefined ? 0 : val;
        this.height = height === undefined ? 0 : height; 
        this.left = left === undefined ? null : left; 
        this.right = right === undefined ? null : right; 
    }
}

/* AVL tree node */
class TreeNode {
  int val;         // Node value
  int height;      // Node height
  TreeNode? left;  // Left child
  TreeNode? right; // Right child
  TreeNode(this.val, [this.height = 0, this.left, this.right]);
}

use std::rc::Rc;
use std::cell::RefCell;

/* AVL tree node */
struct TreeNode {
    val: i32,                               // Node value
    height: i32,                            // Node height
    left: Option<Rc<RefCell<TreeNode>>>,    // Left child
    right: Option<Rc<RefCell<TreeNode>>>,   // Right child
}

impl TreeNode {
    /* Constructor */
    fn new(val: i32) -> Rc<RefCell<Self>> {
        Rc::new(RefCell::new(Self {
            val,
            height: 0,
            left: None,
            right: None
        }))
    }
}

/* AVL tree node */
TreeNode struct TreeNode {
    int val;
    int height;
    struct TreeNode *left;
    struct TreeNode *right;
} TreeNode;

/* Constructor */
TreeNode *newTreeNode(int val) {
    TreeNode *node;

    node = (TreeNode *)malloc(sizeof(TreeNode));
    node->val = val;
    node->height = 0;
    node->left = NULL;
    node->right = NULL;
    return node;
}

/* AVL tree node */
class TreeNode(val _val: Int) {  // Node value
    val height: Int = 0          // Node height
    val left: TreeNode? = null   // Left child
    val right: TreeNode? = null  // Right child
}

The "node height" refers to the distance from that node to its farthest leaf node, i.e., the number of "edges" passed. It is important to note that the height of a leaf node is \(0\), and the height of a null node is \(-1\). We will create two utility functions for getting and updating the height of a node:

avl_tree.py
def height(self, node: TreeNode | None) -> int:
    """获取节点高度"""
    # 空节点高度为 -1 ，叶节点高度为 0
    if node is not None:
        return node.height
    return -1

def update_height(self, node: TreeNode | None):
    """更新节点高度"""
    # 节点高度等于最高子树高度 + 1
    node.height = max([self.height(node.left), self.height(node.right)]) + 1
avl_tree.cpp
/* 获取节点高度 */
int height(TreeNode *node) {
    // 空节点高度为 -1 ，叶节点高度为 0
    return node == nullptr ? -1 : node->height;
}

/* 更新节点高度 */
void updateHeight(TreeNode *node) {
    // 节点高度等于最高子树高度 + 1
    node->height = max(height(node->left), height(node->right)) + 1;
}
avl_tree.java
/* 获取节点高度 */
int height(TreeNode node) {
    // 空节点高度为 -1 ，叶节点高度为 0
    return node == null ? -1 : node.height;
}

/* 更新节点高度 */
void updateHeight(TreeNode node) {
    // 节点高度等于最高子树高度 + 1
    node.height = Math.max(height(node.left), height(node.right)) + 1;
}
avl_tree.cs
/* 获取节点高度 */
int Height(TreeNode? node) {
    // 空节点高度为 -1 ，叶节点高度为 0
    return node == null ? -1 : node.height;
}

/* 更新节点高度 */
void UpdateHeight(TreeNode node) {
    // 节点高度等于最高子树高度 + 1
    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;
}
avl_tree.go
/* 获取节点高度 */
func (t *aVLTree) height(node *TreeNode) int {
    // 空节点高度为 -1 ，叶节点高度为 0
    if node != nil {
        return node.Height
    }
    return -1
}

/* 更新节点高度 */
func (t *aVLTree) updateHeight(node *TreeNode) {
    lh := t.height(node.Left)
    rh := t.height(node.Right)
    // 节点高度等于最高子树高度 + 1
    if lh > rh {
        node.Height = lh + 1
    } else {
        node.Height = rh + 1
    }
}
avl_tree.swift
/* 获取节点高度 */
func height(node: TreeNode?) -> Int {
    // 空节点高度为 -1 ，叶节点高度为 0
    node?.height ?? -1
}

/* 更新节点高度 */
func updateHeight(node: TreeNode?) {
    // 节点高度等于最高子树高度 + 1
    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1
}
avl_tree.js
/* 获取节点高度 */
height(node) {
    // 空节点高度为 -1 ，叶节点高度为 0
    return node === null ? -1 : node.height;
}

/* 更新节点高度 */
#updateHeight(node) {
    // 节点高度等于最高子树高度 + 1
    node.height =
        Math.max(this.height(node.left), this.height(node.right)) + 1;
}
avl_tree.ts
/* 获取节点高度 */
height(node: TreeNode): number {
    // 空节点高度为 -1 ，叶节点高度为 0
    return node === null ? -1 : node.height;
}

/* 更新节点高度 */
updateHeight(node: TreeNode): void {
    // 节点高度等于最高子树高度 + 1
    node.height =
        Math.max(this.height(node.left), this.height(node.right)) + 1;
}
avl_tree.dart
/* 获取节点高度 */
int height(TreeNode? node) {
  // 空节点高度为 -1 ，叶节点高度为 0
  return node == null ? -1 : node.height;
}

/* 更新节点高度 */
void updateHeight(TreeNode? node) {
  // 节点高度等于最高子树高度 + 1
  node!.height = max(height(node.left), height(node.right)) + 1;
}
avl_tree.rs
/* 获取节点高度 */
fn height(node: OptionTreeNodeRc) -> i32 {
    // 空节点高度为 -1 ，叶节点高度为 0
    match node {
        Some(node) => node.borrow().height,
        None => -1,
    }
}

/* 更新节点高度 */
fn update_height(node: OptionTreeNodeRc) {
    if let Some(node) = node {
        let left = node.borrow().left.clone();
        let right = node.borrow().right.clone();
        // 节点高度等于最高子树高度 + 1
        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;
    }
}
avl_tree.c
/* 获取节点高度 */
int height(TreeNode *node) {
    // 空节点高度为 -1 ，叶节点高度为 0
    if (node != NULL) {
        return node->height;
    }
    return -1;
}

/* 更新节点高度 */
void updateHeight(TreeNode *node) {
    int lh = height(node->left);
    int rh = height(node->right);
    // 节点高度等于最高子树高度 + 1
    if (lh > rh) {
        node->height = lh + 1;
    } else {
        node->height = rh + 1;
    }
}
avl_tree.kt
/* 获取节点高度 */
fun height(node: TreeNode?): Int {
    // 空节点高度为 -1 ，叶节点高度为 0
    return node?.height ?: -1
}

/* 更新节点高度 */
fun updateHeight(node: TreeNode?) {
    // 节点高度等于最高子树高度 + 1
    node?.height = (max(height(node?.left).toDouble(), height(node?.right).toDouble()) + 1).toInt()
}
avl_tree.rb
[class]{AVLTree}-[func]{height}

[class]{AVLTree}-[func]{update_height}
avl_tree.zig
// 获取节点高度
fn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {
    _ = self;
    // 空节点高度为 -1 ，叶节点高度为 0
    return if (node == null) -1 else node.?.height;
}

// 更新节点高度
fn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {
    // 节点高度等于最高子树高度 + 1
    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;
}

2.   Node balance factor

The "balance factor" of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as \(0\). We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:

avl_tree.py
def balance_factor(self, node: TreeNode | None) -> int:
    """获取平衡因子"""
    # 空节点平衡因子为 0
    if node is None:
        return 0
    # 节点平衡因子 = 左子树高度 - 右子树高度
    return self.height(node.left) - self.height(node.right)
avl_tree.cpp
/* 获取平衡因子 */
int balanceFactor(TreeNode *node) {
    // 空节点平衡因子为 0
    if (node == nullptr)
        return 0;
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return height(node->left) - height(node->right);
}
avl_tree.java
/* 获取平衡因子 */
int balanceFactor(TreeNode node) {
    // 空节点平衡因子为 0
    if (node == null)
        return 0;
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return height(node.left) - height(node.right);
}
avl_tree.cs
/* 获取平衡因子 */
int BalanceFactor(TreeNode? node) {
    // 空节点平衡因子为 0
    if (node == null) return 0;
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return Height(node.left) - Height(node.right);
}
avl_tree.go
/* 获取平衡因子 */
func (t *aVLTree) balanceFactor(node *TreeNode) int {
    // 空节点平衡因子为 0
    if node == nil {
        return 0
    }
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return t.height(node.Left) - t.height(node.Right)
}
avl_tree.swift
/* 获取平衡因子 */
func balanceFactor(node: TreeNode?) -> Int {
    // 空节点平衡因子为 0
    guard let node = node else { return 0 }
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return height(node: node.left) - height(node: node.right)
}
avl_tree.js
/* 获取平衡因子 */
balanceFactor(node) {
    // 空节点平衡因子为 0
    if (node === null) return 0;
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return this.height(node.left) - this.height(node.right);
}
avl_tree.ts
/* 获取平衡因子 */
balanceFactor(node: TreeNode): number {
    // 空节点平衡因子为 0
    if (node === null) return 0;
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return this.height(node.left) - this.height(node.right);
}
avl_tree.dart
/* 获取平衡因子 */
int balanceFactor(TreeNode? node) {
  // 空节点平衡因子为 0
  if (node == null) return 0;
  // 节点平衡因子 = 左子树高度 - 右子树高度
  return height(node.left) - height(node.right);
}
avl_tree.rs
/* 获取平衡因子 */
fn balance_factor(node: OptionTreeNodeRc) -> i32 {
    match node {
        // 空节点平衡因子为 0
        None => 0,
        // 节点平衡因子 = 左子树高度 - 右子树高度
        Some(node) => {
            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())
        }
    }
}
avl_tree.c
/* 获取平衡因子 */
int balanceFactor(TreeNode *node) {
    // 空节点平衡因子为 0
    if (node == NULL) {
        return 0;
    }
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return height(node->left) - height(node->right);
}
avl_tree.kt
/* 获取平衡因子 */
fun balanceFactor(node: TreeNode?): Int {
    // 空节点平衡因子为 0
    if (node == null) return 0
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return height(node.left) - height(node.right)
}
avl_tree.rb
[class]{AVLTree}-[func]{balance_factor}
avl_tree.zig
// 获取平衡因子
fn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {
    // 空节点平衡因子为 0
    if (node == null) return 0;
    // 节点平衡因子 = 左子树高度 - 右子树高度
    return self.height(node.?.left) - self.height(node.?.right);
}

Note

Let the balance factor be \(f\), then the balance factor of any node in an AVL tree satisfies \(-1 \le f \le 1\).

7.5.2   Rotations in AVL trees

The characteristic feature of an AVL tree is the "rotation" operation, which can restore balance to an unbalanced node without affecting the in-order traversal sequence of the binary tree. In other words, the rotation operation can maintain the property of a "binary search tree" while also turning the tree back into a "balanced binary tree".

We call nodes with an absolute balance factor \(> 1\) "unbalanced nodes". Depending on the type of imbalance, there are four kinds of rotations: right rotation, left rotation, right-left rotation, and left-right rotation. Below, we detail these rotation operations.

1.   Right rotation

As shown in the Figure 7-26 , the first unbalanced node from the bottom up in the binary tree is "node 3". Focusing on the subtree with this unbalanced node as the root, denoted as node, and its left child as child, perform a "right rotation". After the right rotation, the subtree is balanced again while still maintaining the properties of a binary search tree.

Steps of right rotation

avltree_right_rotate_step2

avltree_right_rotate_step3

avltree_right_rotate_step4

Figure 7-26   Steps of right rotation

As shown in the Figure 7-27 , when the child node has a right child (denoted as grand_child), a step needs to be added in the right rotation: set grand_child as the left child of node.

Right rotation with grand_child

Figure 7-27   Right rotation with grand_child

"Right rotation" is a figurative term; in practice, it is achieved by modifying node pointers, as shown in the following code:

avl_tree.py
def right_rotate(self, node: TreeNode | None) -> TreeNode | None:
    """右旋操作"""
    child = node.left
    grand_child = child.right
    # 以 child 为原点，将 node 向右旋转
    child.right = node
    node.left = grand_child
    # 更新节点高度
    self.update_height(node)
    self.update_height(child)
    # 返回旋转后子树的根节点
    return child
avl_tree.cpp
/* 右旋操作 */
TreeNode *rightRotate(TreeNode *node) {
    TreeNode *child = node->left;
    TreeNode *grandChild = child->right;
    // 以 child 为原点，将 node 向右旋转
    child->right = node;
    node->left = grandChild;
    // 更新节点高度
    updateHeight(node);
    updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.java
/* 右旋操作 */
TreeNode rightRotate(TreeNode node) {
    TreeNode child = node.left;
    TreeNode grandChild = child.right;
    // 以 child 为原点，将 node 向右旋转
    child.right = node;
    node.left = grandChild;
    // 更新节点高度
    updateHeight(node);
    updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.cs
/* 右旋操作 */
TreeNode? RightRotate(TreeNode? node) {
    TreeNode? child = node?.left;
    TreeNode? grandChild = child?.right;
    // 以 child 为原点，将 node 向右旋转
    child.right = node;
    node.left = grandChild;
    // 更新节点高度
    UpdateHeight(node);
    UpdateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.go
/* 右旋操作 */
func (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {
    child := node.Left
    grandChild := child.Right
    // 以 child 为原点，将 node 向右旋转
    child.Right = node
    node.Left = grandChild
    // 更新节点高度
    t.updateHeight(node)
    t.updateHeight(child)
    // 返回旋转后子树的根节点
    return child
}
avl_tree.swift
/* 右旋操作 */
func rightRotate(node: TreeNode?) -> TreeNode? {
    let child = node?.left
    let grandChild = child?.right
    // 以 child 为原点，将 node 向右旋转
    child?.right = node
    node?.left = grandChild
    // 更新节点高度
    updateHeight(node: node)
    updateHeight(node: child)
    // 返回旋转后子树的根节点
    return child
}
avl_tree.js
/* 右旋操作 */
#rightRotate(node) {
    const child = node.left;
    const grandChild = child.right;
    // 以 child 为原点，将 node 向右旋转
    child.right = node;
    node.left = grandChild;
    // 更新节点高度
    this.#updateHeight(node);
    this.#updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.ts
/* 右旋操作 */
rightRotate(node: TreeNode): TreeNode {
    const child = node.left;
    const grandChild = child.right;
    // 以 child 为原点，将 node 向右旋转
    child.right = node;
    node.left = grandChild;
    // 更新节点高度
    this.updateHeight(node);
    this.updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.dart
/* 右旋操作 */
TreeNode? rightRotate(TreeNode? node) {
  TreeNode? child = node!.left;
  TreeNode? grandChild = child!.right;
  // 以 child 为原点，将 node 向右旋转
  child.right = node;
  node.left = grandChild;
  // 更新节点高度
  updateHeight(node);
  updateHeight(child);
  // 返回旋转后子树的根节点
  return child;
}
avl_tree.rs
/* 右旋操作 */
fn right_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {
    match node {
        Some(node) => {
            let child = node.borrow().left.clone().unwrap();
            let grand_child = child.borrow().right.clone();
            // 以 child 为原点，将 node 向右旋转
            child.borrow_mut().right = Some(node.clone());
            node.borrow_mut().left = grand_child;
            // 更新节点高度
            Self::update_height(Some(node));
            Self::update_height(Some(child.clone()));
            // 返回旋转后子树的根节点
            Some(child)
        }
        None => None,
    }
}
avl_tree.c
/* 右旋操作 */
TreeNode *rightRotate(TreeNode *node) {
    TreeNode *child, *grandChild;
    child = node->left;
    grandChild = child->right;
    // 以 child 为原点，将 node 向右旋转
    child->right = node;
    node->left = grandChild;
    // 更新节点高度
    updateHeight(node);
    updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.kt
/* 右旋操作 */
fun rightRotate(node: TreeNode?): TreeNode {
    val child = node!!.left
    val grandChild = child!!.right
    // 以 child 为原点，将 node 向右旋转
    child.right = node
    node.left = grandChild
    // 更新节点高度
    updateHeight(node)
    updateHeight(child)
    // 返回旋转后子树的根节点
    return child
}
avl_tree.rb
[class]{AVLTree}-[func]{right_rotate}
avl_tree.zig
// 右旋操作
fn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {
    var child = node.?.left;
    var grandChild = child.?.right;
    // 以 child 为原点，将 node 向右旋转
    child.?.right = node;
    node.?.left = grandChild;
    // 更新节点高度
    self.updateHeight(node);
    self.updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}

2.   Left rotation

Correspondingly, if considering the "mirror" of the above unbalanced binary tree, the "left rotation" operation shown in the Figure 7-28 needs to be performed.

Left rotation operation

Figure 7-28   Left rotation operation

Similarly, as shown in the Figure 7-29 , when the child node has a left child (denoted as grand_child), a step needs to be added in the left rotation: set grand_child as the right child of node.

Left rotation with grand_child

Figure 7-29   Left rotation with grand_child

It can be observed that the right and left rotation operations are logically symmetrical, and they solve two symmetrical types of imbalance. Based on symmetry, by replacing all left with right, and all right with left in the implementation code of right rotation, we can get the implementation code for left rotation:

avl_tree.py
def left_rotate(self, node: TreeNode | None) -> TreeNode | None:
    """左旋操作"""
    child = node.right
    grand_child = child.left
    # 以 child 为原点，将 node 向左旋转
    child.left = node
    node.right = grand_child
    # 更新节点高度
    self.update_height(node)
    self.update_height(child)
    # 返回旋转后子树的根节点
    return child
avl_tree.cpp
/* 左旋操作 */
TreeNode *leftRotate(TreeNode *node) {
    TreeNode *child = node->right;
    TreeNode *grandChild = child->left;
    // 以 child 为原点，将 node 向左旋转
    child->left = node;
    node->right = grandChild;
    // 更新节点高度
    updateHeight(node);
    updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.java
/* 左旋操作 */
TreeNode leftRotate(TreeNode node) {
    TreeNode child = node.right;
    TreeNode grandChild = child.left;
    // 以 child 为原点，将 node 向左旋转
    child.left = node;
    node.right = grandChild;
    // 更新节点高度
    updateHeight(node);
    updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.cs
/* 左旋操作 */
TreeNode? LeftRotate(TreeNode? node) {
    TreeNode? child = node?.right;
    TreeNode? grandChild = child?.left;
    // 以 child 为原点，将 node 向左旋转
    child.left = node;
    node.right = grandChild;
    // 更新节点高度
    UpdateHeight(node);
    UpdateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.go
/* 左旋操作 */
func (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {
    child := node.Right
    grandChild := child.Left
    // 以 child 为原点，将 node 向左旋转
    child.Left = node
    node.Right = grandChild
    // 更新节点高度
    t.updateHeight(node)
    t.updateHeight(child)
    // 返回旋转后子树的根节点
    return child
}
avl_tree.swift
/* 左旋操作 */
func leftRotate(node: TreeNode?) -> TreeNode? {
    let child = node?.right
    let grandChild = child?.left
    // 以 child 为原点，将 node 向左旋转
    child?.left = node
    node?.right = grandChild
    // 更新节点高度
    updateHeight(node: node)
    updateHeight(node: child)
    // 返回旋转后子树的根节点
    return child
}
avl_tree.js
/* 左旋操作 */
#leftRotate(node) {
    const child = node.right;
    const grandChild = child.left;
    // 以 child 为原点，将 node 向左旋转
    child.left = node;
    node.right = grandChild;
    // 更新节点高度
    this.#updateHeight(node);
    this.#updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.ts
/* 左旋操作 */
leftRotate(node: TreeNode): TreeNode {
    const child = node.right;
    const grandChild = child.left;
    // 以 child 为原点，将 node 向左旋转
    child.left = node;
    node.right = grandChild;
    // 更新节点高度
    this.updateHeight(node);
    this.updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.dart
/* 左旋操作 */
TreeNode? leftRotate(TreeNode? node) {
  TreeNode? child = node!.right;
  TreeNode? grandChild = child!.left;
  // 以 child 为原点，将 node 向左旋转
  child.left = node;
  node.right = grandChild;
  // 更新节点高度
  updateHeight(node);
  updateHeight(child);
  // 返回旋转后子树的根节点
  return child;
}
avl_tree.rs
/* 左旋操作 */
fn left_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {
    match node {
        Some(node) => {
            let child = node.borrow().right.clone().unwrap();
            let grand_child = child.borrow().left.clone();
            // 以 child 为原点，将 node 向左旋转
            child.borrow_mut().left = Some(node.clone());
            node.borrow_mut().right = grand_child;
            // 更新节点高度
            Self::update_height(Some(node));
            Self::update_height(Some(child.clone()));
            // 返回旋转后子树的根节点
            Some(child)
        }
        None => None,
    }
}
avl_tree.c
/* 左旋操作 */
TreeNode *leftRotate(TreeNode *node) {
    TreeNode *child, *grandChild;
    child = node->right;
    grandChild = child->left;
    // 以 child 为原点，将 node 向左旋转
    child->left = node;
    node->right = grandChild;
    // 更新节点高度
    updateHeight(node);
    updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}
avl_tree.kt
/* 左旋操作 */
fun leftRotate(node: TreeNode?): TreeNode {
    val child = node!!.right
    val grandChild = child!!.left
    // 以 child 为原点，将 node 向左旋转
    child.left = node
    node.right = grandChild
    // 更新节点高度
    updateHeight(node)
    updateHeight(child)
    // 返回旋转后子树的根节点
    return child
}
avl_tree.rb
[class]{AVLTree}-[func]{left_rotate}
avl_tree.zig
// 左旋操作
fn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {
    var child = node.?.right;
    var grandChild = child.?.left;
    // 以 child 为原点，将 node 向左旋转
    child.?.left = node;
    node.?.right = grandChild;
    // 更新节点高度
    self.updateHeight(node);
    self.updateHeight(child);
    // 返回旋转后子树的根节点
    return child;
}

3.   Right-left rotation

For the unbalanced node 3 shown in the Figure 7-30 , using either left or right rotation alone cannot restore balance to the subtree. In this case, a "left rotation" needs to be performed on child first, followed by a "right rotation" on node.

Right-left rotation

Figure 7-30   Right-left rotation

4.   Left-right rotation

As shown in the Figure 7-31 , for the mirror case of the above unbalanced binary tree, a "right rotation" needs to be performed on child first, followed by a "left rotation" on node.

Left-right rotation

Figure 7-31   Left-right rotation

5.   Choice of rotation

The four kinds of imbalances shown in the Figure 7-32 correspond to the cases described above, respectively requiring right rotation, left-right rotation, right-left rotation, and left rotation.

The four rotation cases of AVL tree

Figure 7-32   The four rotation cases of AVL tree

As shown in the Table 7-3 , we determine which of the above cases an unbalanced node belongs to by judging the sign of the balance factor of the unbalanced node and its higher-side child's balance factor.

Table 7-3   Conditions for Choosing Among the Four Rotation Cases

Balance factor of unbalanced node Balance factor of child node Rotation method to use
\(> 1\) (Left-leaning tree) \(\geq 0\) Right rotation
\(> 1\) (Left-leaning tree) \(<0\) Left rotation then right rotation
\(< -1\) (Right-leaning tree) \(\leq 0\) Left rotation
\(< -1\) (Right-leaning tree) \(>0\) Right rotation then left rotation

For convenience, we encapsulate the rotation operations into a function. With this function, we can perform rotations on various kinds of imbalances, restoring balance to unbalanced nodes. The code is as follows:

avl_tree.py
def rotate(self, node: TreeNode | None) -> TreeNode | None:
    """执行旋转操作，使该子树重新恢复平衡"""
    # 获取节点 node 的平衡因子
    balance_factor = self.balance_factor(node)
    # 左偏树
    if balance_factor > 1:
        if self.balance_factor(node.left) >= 0:
            # 右旋
            return self.right_rotate(node)
        else:
            # 先左旋后右旋
            node.left = self.left_rotate(node.left)
            return self.right_rotate(node)
    # 右偏树
    elif balance_factor < -1:
        if self.balance_factor(node.right) <= 0:
            # 左旋
            return self.left_rotate(node)
        else:
            # 先右旋后左旋
            node.right = self.right_rotate(node.right)
            return self.left_rotate(node)
    # 平衡树，无须旋转，直接返回
    return node
avl_tree.cpp
/* 执行旋转操作，使该子树重新恢复平衡 */
TreeNode *rotate(TreeNode *node) {
    // 获取节点 node 的平衡因子
    int _balanceFactor = balanceFactor(node);
    // 左偏树
    if (_balanceFactor > 1) {
        if (balanceFactor(node->left) >= 0) {
            // 右旋
            return rightRotate(node);
        } else {
            // 先左旋后右旋
            node->left = leftRotate(node->left);
            return rightRotate(node);
        }
    }
    // 右偏树
    if (_balanceFactor < -1) {
        if (balanceFactor(node->right) <= 0) {
            // 左旋
            return leftRotate(node);
        } else {
            // 先右旋后左旋
            node->right = rightRotate(node->right);
            return leftRotate(node);
        }
    }
    // 平衡树，无须旋转，直接返回
    return node;
}
avl_tree.java
/* 执行旋转操作，使该子树重新恢复平衡 */
TreeNode rotate(TreeNode node) {
    // 获取节点 node 的平衡因子
    int balanceFactor = balanceFactor(node);
    // 左偏树
    if (balanceFactor > 1) {
        if (balanceFactor(node.left) >= 0) {
            // 右旋
            return rightRotate(node);
        } else {
            // 先左旋后右旋
            node.left = leftRotate(node.left);
            return rightRotate(node);
        }
    }
    // 右偏树
    if (balanceFactor < -1) {
        if (balanceFactor(node.right) <= 0) {
            // 左旋
            return leftRotate(node);
        } else {
            // 先右旋后左旋
            node.right = rightRotate(node.right);
            return leftRotate(node);
        }
    }
    // 平衡树，无须旋转，直接返回
    return node;
}
avl_tree.cs
/* 执行旋转操作，使该子树重新恢复平衡 */
TreeNode? Rotate(TreeNode? node) {
    // 获取节点 node 的平衡因子
    int balanceFactorInt = BalanceFactor(node);
    // 左偏树
    if (balanceFactorInt > 1) {
        if (BalanceFactor(node?.left) >= 0) {
            // 右旋
            return RightRotate(node);
        } else {
            // 先左旋后右旋
            node!.left = LeftRotate(node!.left);
            return RightRotate(node);
        }
    }
    // 右偏树
    if (balanceFactorInt < -1) {
        if (BalanceFactor(node?.right) <= 0) {
            // 左旋
            return LeftRotate(node);
        } else {
            // 先右旋后左旋
            node!.right = RightRotate(node!.right);
            return LeftRotate(node);
        }
    }
    // 平衡树，无须旋转，直接返回
    return node;
}
avl_tree.go
/* 执行旋转操作，使该子树重新恢复平衡 */
func (t *aVLTree) rotate(node *TreeNode) *TreeNode {
    // 获取节点 node 的平衡因子
    // Go 推荐短变量，这里 bf 指代 t.balanceFactor
    bf := t.balanceFactor(node)
    // 左偏树
    if bf > 1 {
        if t.balanceFactor(node.Left) >= 0 {
            // 右旋
            return t.rightRotate(node)
        } else {
            // 先左旋后右旋
            node.Left = t.leftRotate(node.Left)
            return t.rightRotate(node)
        }
    }
    // 右偏树
    if bf < -1 {
        if t.balanceFactor(node.Right) <= 0 {
            // 左旋
            return t.leftRotate(node)
        } else {
            // 先右旋后左旋
            node.Right = t.rightRotate(node.Right)
            return t.leftRotate(node)
        }
    }
    // 平衡树，无须旋转，直接返回
    return node
}
avl_tree.swift
/* 执行旋转操作，使该子树重新恢复平衡 */
func rotate(node: TreeNode?) -> TreeNode? {
    // 获取节点 node 的平衡因子
    let balanceFactor = balanceFactor(node: node)
    // 左偏树
    if balanceFactor > 1 {
        if self.balanceFactor(node: node?.left) >= 0 {
            // 右旋
            return rightRotate(node: node)
        } else {
            // 先左旋后右旋
            node?.left = leftRotate(node: node?.left)
            return rightRotate(node: node)
        }
    }
    // 右偏树
    if balanceFactor < -1 {
        if self.balanceFactor(node: node?.right) <= 0 {
            // 左旋
            return leftRotate(node: node)
        } else {
            // 先右旋后左旋
            node?.right = rightRotate(node: node?.right)
            return leftRotate(node: node)
        }
    }
    // 平衡树，无须旋转，直接返回
    return node
}
avl_tree.js
/* 执行旋转操作，使该子树重新恢复平衡 */
#rotate(node) {
    // 获取节点 node 的平衡因子
    const balanceFactor = this.balanceFactor(node);
    // 左偏树
    if (balanceFactor > 1) {
        if (this.balanceFactor(node.left) >= 0) {
            // 右旋
            return this.#rightRotate(node);
        } else {
            // 先左旋后右旋
            node.left = this.#leftRotate(node.left);
            return this.#rightRotate(node);
        }
    }
    // 右偏树
    if (balanceFactor < -1) {
        if (this.balanceFactor(node.right) <= 0) {
            // 左旋
            return this.#leftRotate(node);
        } else {
            // 先右旋后左旋
            node.right = this.#rightRotate(node.right);
            return this.#leftRotate(node);
        }
    }
    // 平衡树，无须旋转，直接返回
    return node;
}
avl_tree.ts
/* 执行旋转操作，使该子树重新恢复平衡 */
rotate(node: TreeNode): TreeNode {
    // 获取节点 node 的平衡因子
    const balanceFactor = this.balanceFactor(node);
    // 左偏树
    if (balanceFactor > 1) {
        if (this.balanceFactor(node.left) >= 0) {
            // 右旋
            return this.rightRotate(node);
        } else {
            // 先左旋后右旋
            node.left = this.leftRotate(node.left);
            return this.rightRotate(node);
        }
    }
    // 右偏树
    if (balanceFactor < -1) {
        if (this.balanceFactor(node.right) <= 0) {
            // 左旋
            return this.leftRotate(node);
        } else {
            // 先右旋后左旋
            node.right = this.rightRotate(node.right);
            return this.leftRotate(node);
        }
    }
    // 平衡树，无须旋转，直接返回
    return node;
}
avl_tree.dart
/* 执行旋转操作，使该子树重新恢复平衡 */
TreeNode? rotate(TreeNode? node) {
  // 获取节点 node 的平衡因子
  int factor = balanceFactor(node);
  // 左偏树
  if (factor > 1) {
    if (balanceFactor(node!.left) >= 0) {
      // 右旋
      return rightRotate(node);
    } else {
      // 先左旋后右旋
      node.left = leftRotate(node.left);
      return rightRotate(node);
    }
  }
  // 右偏树
  if (factor < -1) {
    if (balanceFactor(node!.right) <= 0) {
      // 左旋
      return leftRotate(node);
    } else {
      // 先右旋后左旋
      node.right = rightRotate(node.right);
      return leftRotate(node);
    }
  }
  // 平衡树，无须旋转，直接返回
  return node;
}
avl_tree.rs
/* 执行旋转操作，使该子树重新恢复平衡 */
fn rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {
    // 获取节点 node 的平衡因子
    let balance_factor = Self::balance_factor(node.clone());
    // 左偏树
    if balance_factor > 1 {
        let node = node.unwrap();
        if Self::balance_factor(node.borrow().left.clone()) >= 0 {
            // 右旋
            Self::right_rotate(Some(node))
        } else {
            // 先左旋后右旋
            let left = node.borrow().left.clone();
            node.borrow_mut().left = Self::left_rotate(left);
            Self::right_rotate(Some(node))
        }
    }
    // 右偏树
    else if balance_factor < -1 {
        let node = node.unwrap();
        if Self::balance_factor(node.borrow().right.clone()) <= 0 {
            // 左旋
            Self::left_rotate(Some(node))
        } else {
            // 先右旋后左旋
            let right = node.borrow().right.clone();
            node.borrow_mut().right = Self::right_rotate(right);
            Self::left_rotate(Some(node))
        }
    } else {
        // 平衡树，无须旋转，直接返回
        node
    }
}
avl_tree.c
/* 执行旋转操作，使该子树重新恢复平衡 */
TreeNode *rotate(TreeNode *node) {
    // 获取节点 node 的平衡因子
    int bf = balanceFactor(node);
    // 左偏树
    if (bf > 1) {
        if (balanceFactor(node->left) >= 0) {
            // 右旋
            return rightRotate(node);
        } else {
            // 先左旋后右旋
            node->left = leftRotate(node->left);
            return rightRotate(node);
        }
    }
    // 右偏树
    if (bf < -1) {
        if (balanceFactor(node->right) <= 0) {
            // 左旋
            return leftRotate(node);
        } else {
            // 先右旋后左旋
            node->right = rightRotate(node->right);
            return leftRotate(node);
        }
    }
    // 平衡树，无须旋转，直接返回
    return node;
}
avl_tree.kt
/* 执行旋转操作，使该子树重新恢复平衡 */
fun rotate(node: TreeNode): TreeNode {
    // 获取节点 node 的平衡因子
    val balanceFactor = balanceFactor(node)
    // 左偏树
    if (balanceFactor > 1) {
        if (balanceFactor(node.left) >= 0) {
            // 右旋
            return rightRotate(node)
        } else {
            // 先左旋后右旋
            node.left = leftRotate(node.left)
            return rightRotate(node)
        }
    }
    // 右偏树
    if (balanceFactor < -1) {
        if (balanceFactor(node.right) <= 0) {
            // 左旋
            return leftRotate(node)
        } else {
            // 先右旋后左旋
            node.right = rightRotate(node.right)
            return leftRotate(node)
        }
    }
    // 平衡树，无须旋转，直接返回
    return node
}
avl_tree.rb
[class]{AVLTree}-[func]{rotate}
avl_tree.zig
// 执行旋转操作，使该子树重新恢复平衡
fn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {
    // 获取节点 node 的平衡因子
    var balance_factor = self.balanceFactor(node);
    // 左偏树
    if (balance_factor > 1) {
        if (self.balanceFactor(node.?.left) >= 0) {
            // 右旋
            return self.rightRotate(node);
        } else {
            // 先左旋后右旋
            node.?.left = self.leftRotate(node.?.left);
            return self.rightRotate(node);
        }
    }
    // 右偏树
    if (balance_factor < -1) {
        if (self.balanceFactor(node.?.right) <= 0) {
            // 左旋
            return self.leftRotate(node);
        } else {
            // 先右旋后左旋
            node.?.right = self.rightRotate(node.?.right);
            return self.leftRotate(node);
        }
    }
    // 平衡树，无须旋转，直接返回
    return node;
}

7.5.3   Common operations in AVL trees

1.   Node insertion

The node insertion operation in AVL trees is similar to that in binary search trees. The only difference is that after inserting a node in an AVL tree, a series of unbalanced nodes may appear along the path from that node to the root node. Therefore, we need to start from this node and perform rotation operations upwards to restore balance to all unbalanced nodes. The code is as follows:

avl_tree.py
def insert(self, val):
    """插入节点"""
    self._root = self.insert_helper(self._root, val)

def insert_helper(self, node: TreeNode | None, val: int) -> TreeNode:
    """递归插入节点（辅助方法）"""
    if node is None:
        return TreeNode(val)
    # 1. 查找插入位置并插入节点
    if val < node.val:
        node.left = self.insert_helper(node.left, val)
    elif val > node.val:
        node.right = self.insert_helper(node.right, val)
    else:
        # 重复节点不插入，直接返回
        return node
    # 更新节点高度
    self.update_height(node)
    # 2. 执行旋转操作，使该子树重新恢复平衡
    return self.rotate(node)
avl_tree.cpp
/* 插入节点 */
void insert(int val) {
    root = insertHelper(root, val);
}

/* 递归插入节点（辅助方法） */
TreeNode *insertHelper(TreeNode *node, int val) {
    if (node == nullptr)
        return new TreeNode(val);
    /* 1. 查找插入位置并插入节点 */
    if (val < node->val)
        node->left = insertHelper(node->left, val);
    else if (val > node->val)
        node->right = insertHelper(node->right, val);
    else
        return node;    // 重复节点不插入，直接返回
    updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.java
/* 插入节点 */
void insert(int val) {
    root = insertHelper(root, val);
}

/* 递归插入节点（辅助方法） */
TreeNode insertHelper(TreeNode node, int val) {
    if (node == null)
        return new TreeNode(val);
    /* 1. 查找插入位置并插入节点 */
    if (val < node.val)
        node.left = insertHelper(node.left, val);
    else if (val > node.val)
        node.right = insertHelper(node.right, val);
    else
        return node; // 重复节点不插入，直接返回
    updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.cs
/* 插入节点 */
void Insert(int val) {
    root = InsertHelper(root, val);
}

/* 递归插入节点（辅助方法） */
TreeNode? InsertHelper(TreeNode? node, int val) {
    if (node == null) return new TreeNode(val);
    /* 1. 查找插入位置并插入节点 */
    if (val < node.val)
        node.left = InsertHelper(node.left, val);
    else if (val > node.val)
        node.right = InsertHelper(node.right, val);
    else
        return node;     // 重复节点不插入，直接返回
    UpdateHeight(node);  // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = Rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.go
/* 插入节点 */
func (t *aVLTree) insert(val int) {
    t.root = t.insertHelper(t.root, val)
}

/* 递归插入节点（辅助函数） */
func (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {
    if node == nil {
        return NewTreeNode(val)
    }
    /* 1. 查找插入位置并插入节点 */
    if val < node.Val.(int) {
        node.Left = t.insertHelper(node.Left, val)
    } else if val > node.Val.(int) {
        node.Right = t.insertHelper(node.Right, val)
    } else {
        // 重复节点不插入，直接返回
        return node
    }
    // 更新节点高度
    t.updateHeight(node)
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = t.rotate(node)
    // 返回子树的根节点
    return node
}
avl_tree.swift
/* 插入节点 */
func insert(val: Int) {
    root = insertHelper(node: root, val: val)
}

/* 递归插入节点（辅助方法） */
func insertHelper(node: TreeNode?, val: Int) -> TreeNode? {
    var node = node
    if node == nil {
        return TreeNode(x: val)
    }
    /* 1. 查找插入位置并插入节点 */
    if val < node!.val {
        node?.left = insertHelper(node: node?.left, val: val)
    } else if val > node!.val {
        node?.right = insertHelper(node: node?.right, val: val)
    } else {
        return node // 重复节点不插入，直接返回
    }
    updateHeight(node: node) // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node: node)
    // 返回子树的根节点
    return node
}
avl_tree.js
/* 插入节点 */
insert(val) {
    this.root = this.#insertHelper(this.root, val);
}

/* 递归插入节点（辅助方法） */
#insertHelper(node, val) {
    if (node === null) return new TreeNode(val);
    /* 1. 查找插入位置并插入节点 */
    if (val < node.val) node.left = this.#insertHelper(node.left, val);
    else if (val > node.val)
        node.right = this.#insertHelper(node.right, val);
    else return node; // 重复节点不插入，直接返回
    this.#updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = this.#rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.ts
/* 插入节点 */
insert(val: number): void {
    this.root = this.insertHelper(this.root, val);
}

/* 递归插入节点（辅助方法） */
insertHelper(node: TreeNode, val: number): TreeNode {
    if (node === null) return new TreeNode(val);
    /* 1. 查找插入位置并插入节点 */
    if (val < node.val) {
        node.left = this.insertHelper(node.left, val);
    } else if (val > node.val) {
        node.right = this.insertHelper(node.right, val);
    } else {
        return node; // 重复节点不插入，直接返回
    }
    this.updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = this.rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.dart
/* 插入节点 */
void insert(int val) {
  root = insertHelper(root, val);
}

/* 递归插入节点（辅助方法） */
TreeNode? insertHelper(TreeNode? node, int val) {
  if (node == null) return TreeNode(val);
  /* 1. 查找插入位置并插入节点 */
  if (val < node.val)
    node.left = insertHelper(node.left, val);
  else if (val > node.val)
    node.right = insertHelper(node.right, val);
  else
    return node; // 重复节点不插入，直接返回
  updateHeight(node); // 更新节点高度
  /* 2. 执行旋转操作，使该子树重新恢复平衡 */
  node = rotate(node);
  // 返回子树的根节点
  return node;
}
avl_tree.rs
/* 插入节点 */
fn insert(&mut self, val: i32) {
    self.root = Self::insert_helper(self.root.clone(), val);
}

/* 递归插入节点（辅助方法） */
fn insert_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {
    match node {
        Some(mut node) => {
            /* 1. 查找插入位置并插入节点 */
            match {
                let node_val = node.borrow().val;
                node_val
            }
            .cmp(&val)
            {
                Ordering::Greater => {
                    let left = node.borrow().left.clone();
                    node.borrow_mut().left = Self::insert_helper(left, val);
                }
                Ordering::Less => {
                    let right = node.borrow().right.clone();
                    node.borrow_mut().right = Self::insert_helper(right, val);
                }
                Ordering::Equal => {
                    return Some(node); // 重复节点不插入，直接返回
                }
            }
            Self::update_height(Some(node.clone())); // 更新节点高度

            /* 2. 执行旋转操作，使该子树重新恢复平衡 */
            node = Self::rotate(Some(node)).unwrap();
            // 返回子树的根节点
            Some(node)
        }
        None => Some(TreeNode::new(val)),
    }
}
avl_tree.c
/* 插入节点 */
void insert(AVLTree *tree, int val) {
    tree->root = insertHelper(tree->root, val);
}

/* 递归插入节点（辅助函数） */
TreeNode *insertHelper(TreeNode *node, int val) {
    if (node == NULL) {
        return newTreeNode(val);
    }
    /* 1. 查找插入位置并插入节点 */
    if (val < node->val) {
        node->left = insertHelper(node->left, val);
    } else if (val > node->val) {
        node->right = insertHelper(node->right, val);
    } else {
        // 重复节点不插入，直接返回
        return node;
    }
    // 更新节点高度
    updateHeight(node);
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.kt
/* 插入节点 */
fun insert(value: Int) {
    root = insertHelper(root, value)
}

/* 递归插入节点（辅助方法） */
fun insertHelper(n: TreeNode?, value: Int): TreeNode {
    if (n == null)
        return TreeNode(value)
    var node = n
    /* 1. 查找插入位置并插入节点 */
    if (value < node.value) node.left = insertHelper(node.left, value)
    else if (value > node.value) node.right = insertHelper(node.right, value)
    else return node // 重复节点不插入，直接返回

    updateHeight(node) // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node)
    // 返回子树的根节点
    return node
}
avl_tree.rb
[class]{AVLTree}-[func]{insert}

[class]{AVLTree}-[func]{insert_helper}
avl_tree.zig
// 插入节点
fn insert(self: *Self, val: T) !void {
    self.root = (try self.insertHelper(self.root, val)).?;
}

// 递归插入节点（辅助方法）
fn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {
    var node = node_;
    if (node == null) {
        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));
        tmp_node.init(val);
        return tmp_node;
    }
    // 1. 查找插入位置并插入节点
    if (val < node.?.val) {
        node.?.left = try self.insertHelper(node.?.left, val);
    } else if (val > node.?.val) {
        node.?.right = try self.insertHelper(node.?.right, val);
    } else {
        return node;            // 重复节点不插入，直接返回
    }
    self.updateHeight(node);    // 更新节点高度
    // 2. 执行旋转操作，使该子树重新恢复平衡
    node = self.rotate(node);
    // 返回子树的根节点
    return node;
}

2.   Node removal

Similarly, based on the method of removing nodes in binary search trees, rotation operations need to be performed from the bottom up to restore balance to all unbalanced nodes. The code is as follows:

avl_tree.py
def remove(self, val: int):
    """删除节点"""
    self._root = self.remove_helper(self._root, val)

def remove_helper(self, node: TreeNode | None, val: int) -> TreeNode | None:
    """递归删除节点（辅助方法）"""
    if node is None:
        return None
    # 1. 查找节点并删除
    if val < node.val:
        node.left = self.remove_helper(node.left, val)
    elif val > node.val:
        node.right = self.remove_helper(node.right, val)
    else:
        if node.left is None or node.right is None:
            child = node.left or node.right
            # 子节点数量 = 0 ，直接删除 node 并返回
            if child is None:
                return None
            # 子节点数量 = 1 ，直接删除 node
            else:
                node = child
        else:
            # 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            temp = node.right
            while temp.left is not None:
                temp = temp.left
            node.right = self.remove_helper(node.right, temp.val)
            node.val = temp.val
    # 更新节点高度
    self.update_height(node)
    # 2. 执行旋转操作，使该子树重新恢复平衡
    return self.rotate(node)
avl_tree.cpp
/* 删除节点 */
void remove(int val) {
    root = removeHelper(root, val);
}

/* 递归删除节点（辅助方法） */
TreeNode *removeHelper(TreeNode *node, int val) {
    if (node == nullptr)
        return nullptr;
    /* 1. 查找节点并删除 */
    if (val < node->val)
        node->left = removeHelper(node->left, val);
    else if (val > node->val)
        node->right = removeHelper(node->right, val);
    else {
        if (node->left == nullptr || node->right == nullptr) {
            TreeNode *child = node->left != nullptr ? node->left : node->right;
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child == nullptr) {
                delete node;
                return nullptr;
            }
            // 子节点数量 = 1 ，直接删除 node
            else {
                delete node;
                node = child;
            }
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            TreeNode *temp = node->right;
            while (temp->left != nullptr) {
                temp = temp->left;
            }
            int tempVal = temp->val;
            node->right = removeHelper(node->right, temp->val);
            node->val = tempVal;
        }
    }
    updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.java
/* 删除节点 */
void remove(int val) {
    root = removeHelper(root, val);
}

/* 递归删除节点（辅助方法） */
TreeNode removeHelper(TreeNode node, int val) {
    if (node == null)
        return null;
    /* 1. 查找节点并删除 */
    if (val < node.val)
        node.left = removeHelper(node.left, val);
    else if (val > node.val)
        node.right = removeHelper(node.right, val);
    else {
        if (node.left == null || node.right == null) {
            TreeNode child = node.left != null ? node.left : node.right;
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child == null)
                return null;
            // 子节点数量 = 1 ，直接删除 node
            else
                node = child;
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            TreeNode temp = node.right;
            while (temp.left != null) {
                temp = temp.left;
            }
            node.right = removeHelper(node.right, temp.val);
            node.val = temp.val;
        }
    }
    updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.cs
/* 删除节点 */
void Remove(int val) {
    root = RemoveHelper(root, val);
}

/* 递归删除节点（辅助方法） */
TreeNode? RemoveHelper(TreeNode? node, int val) {
    if (node == null) return null;
    /* 1. 查找节点并删除 */
    if (val < node.val)
        node.left = RemoveHelper(node.left, val);
    else if (val > node.val)
        node.right = RemoveHelper(node.right, val);
    else {
        if (node.left == null || node.right == null) {
            TreeNode? child = node.left ?? node.right;
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child == null)
                return null;
            // 子节点数量 = 1 ，直接删除 node
            else
                node = child;
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            TreeNode? temp = node.right;
            while (temp.left != null) {
                temp = temp.left;
            }
            node.right = RemoveHelper(node.right, temp.val!.Value);
            node.val = temp.val;
        }
    }
    UpdateHeight(node);  // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = Rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.go
/* 删除节点 */
func (t *aVLTree) remove(val int) {
    t.root = t.removeHelper(t.root, val)
}

/* 递归删除节点（辅助函数） */
func (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {
    if node == nil {
        return nil
    }
    /* 1. 查找节点并删除 */
    if val < node.Val.(int) {
        node.Left = t.removeHelper(node.Left, val)
    } else if val > node.Val.(int) {
        node.Right = t.removeHelper(node.Right, val)
    } else {
        if node.Left == nil || node.Right == nil {
            child := node.Left
            if node.Right != nil {
                child = node.Right
            }
            if child == nil {
                // 子节点数量 = 0 ，直接删除 node 并返回
                return nil
            } else {
                // 子节点数量 = 1 ，直接删除 node
                node = child
            }
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            temp := node.Right
            for temp.Left != nil {
                temp = temp.Left
            }
            node.Right = t.removeHelper(node.Right, temp.Val.(int))
            node.Val = temp.Val
        }
    }
    // 更新节点高度
    t.updateHeight(node)
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = t.rotate(node)
    // 返回子树的根节点
    return node
}
avl_tree.swift
/* 删除节点 */
func remove(val: Int) {
    root = removeHelper(node: root, val: val)
}

/* 递归删除节点（辅助方法） */
func removeHelper(node: TreeNode?, val: Int) -> TreeNode? {
    var node = node
    if node == nil {
        return nil
    }
    /* 1. 查找节点并删除 */
    if val < node!.val {
        node?.left = removeHelper(node: node?.left, val: val)
    } else if val > node!.val {
        node?.right = removeHelper(node: node?.right, val: val)
    } else {
        if node?.left == nil || node?.right == nil {
            let child = node?.left ?? node?.right
            // 子节点数量 = 0 ，直接删除 node 并返回
            if child == nil {
                return nil
            }
            // 子节点数量 = 1 ，直接删除 node
            else {
                node = child
            }
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            var temp = node?.right
            while temp?.left != nil {
                temp = temp?.left
            }
            node?.right = removeHelper(node: node?.right, val: temp!.val)
            node?.val = temp!.val
        }
    }
    updateHeight(node: node) // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node: node)
    // 返回子树的根节点
    return node
}
avl_tree.js
/* 删除节点 */
remove(val) {
    this.root = this.#removeHelper(this.root, val);
}

/* 递归删除节点（辅助方法） */
#removeHelper(node, val) {
    if (node === null) return null;
    /* 1. 查找节点并删除 */
    if (val < node.val) node.left = this.#removeHelper(node.left, val);
    else if (val > node.val)
        node.right = this.#removeHelper(node.right, val);
    else {
        if (node.left === null || node.right === null) {
            const child = node.left !== null ? node.left : node.right;
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child === null) return null;
            // 子节点数量 = 1 ，直接删除 node
            else node = child;
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            let temp = node.right;
            while (temp.left !== null) {
                temp = temp.left;
            }
            node.right = this.#removeHelper(node.right, temp.val);
            node.val = temp.val;
        }
    }
    this.#updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = this.#rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.ts
/* 删除节点 */
remove(val: number): void {
    this.root = this.removeHelper(this.root, val);
}

/* 递归删除节点（辅助方法） */
removeHelper(node: TreeNode, val: number): TreeNode {
    if (node === null) return null;
    /* 1. 查找节点并删除 */
    if (val < node.val) {
        node.left = this.removeHelper(node.left, val);
    } else if (val > node.val) {
        node.right = this.removeHelper(node.right, val);
    } else {
        if (node.left === null || node.right === null) {
            const child = node.left !== null ? node.left : node.right;
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child === null) {
                return null;
            } else {
                // 子节点数量 = 1 ，直接删除 node
                node = child;
            }
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            let temp = node.right;
            while (temp.left !== null) {
                temp = temp.left;
            }
            node.right = this.removeHelper(node.right, temp.val);
            node.val = temp.val;
        }
    }
    this.updateHeight(node); // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = this.rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.dart
/* 删除节点 */
void remove(int val) {
  root = removeHelper(root, val);
}

/* 递归删除节点（辅助方法） */
TreeNode? removeHelper(TreeNode? node, int val) {
  if (node == null) return null;
  /* 1. 查找节点并删除 */
  if (val < node.val)
    node.left = removeHelper(node.left, val);
  else if (val > node.val)
    node.right = removeHelper(node.right, val);
  else {
    if (node.left == null || node.right == null) {
      TreeNode? child = node.left ?? node.right;
      // 子节点数量 = 0 ，直接删除 node 并返回
      if (child == null)
        return null;
      // 子节点数量 = 1 ，直接删除 node
      else
        node = child;
    } else {
      // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
      TreeNode? temp = node.right;
      while (temp!.left != null) {
        temp = temp.left;
      }
      node.right = removeHelper(node.right, temp.val);
      node.val = temp.val;
    }
  }
  updateHeight(node); // 更新节点高度
  /* 2. 执行旋转操作，使该子树重新恢复平衡 */
  node = rotate(node);
  // 返回子树的根节点
  return node;
}
avl_tree.rs
/* 删除节点 */
fn remove(&self, val: i32) {
    Self::remove_helper(self.root.clone(), val);
}

/* 递归删除节点（辅助方法） */
fn remove_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {
    match node {
        Some(mut node) => {
            /* 1. 查找节点并删除 */
            if val < node.borrow().val {
                let left = node.borrow().left.clone();
                node.borrow_mut().left = Self::remove_helper(left, val);
            } else if val > node.borrow().val {
                let right = node.borrow().right.clone();
                node.borrow_mut().right = Self::remove_helper(right, val);
            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {
                let child = if node.borrow().left.is_some() {
                    node.borrow().left.clone()
                } else {
                    node.borrow().right.clone()
                };
                match child {
                    // 子节点数量 = 0 ，直接删除 node 并返回
                    None => {
                        return None;
                    }
                    // 子节点数量 = 1 ，直接删除 node
                    Some(child) => node = child,
                }
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                let mut temp = node.borrow().right.clone().unwrap();
                loop {
                    let temp_left = temp.borrow().left.clone();
                    if temp_left.is_none() {
                        break;
                    }
                    temp = temp_left.unwrap();
                }
                let right = node.borrow().right.clone();
                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);
                node.borrow_mut().val = temp.borrow().val;
            }
            Self::update_height(Some(node.clone())); // 更新节点高度

            /* 2. 执行旋转操作，使该子树重新恢复平衡 */
            node = Self::rotate(Some(node)).unwrap();
            // 返回子树的根节点
            Some(node)
        }
        None => None,
    }
}
avl_tree.c
/* 删除节点 */
// 由于引入了 stdio.h ，此处无法使用 remove 关键词
void removeItem(AVLTree *tree, int val) {
    TreeNode *root = removeHelper(tree->root, val);
}

/* 递归删除节点（辅助函数） */
TreeNode *removeHelper(TreeNode *node, int val) {
    TreeNode *child, *grandChild;
    if (node == NULL) {
        return NULL;
    }
    /* 1. 查找节点并删除 */
    if (val < node->val) {
        node->left = removeHelper(node->left, val);
    } else if (val > node->val) {
        node->right = removeHelper(node->right, val);
    } else {
        if (node->left == NULL || node->right == NULL) {
            child = node->left;
            if (node->right != NULL) {
                child = node->right;
            }
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child == NULL) {
                return NULL;
            } else {
                // 子节点数量 = 1 ，直接删除 node
                node = child;
            }
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            TreeNode *temp = node->right;
            while (temp->left != NULL) {
                temp = temp->left;
            }
            int tempVal = temp->val;
            node->right = removeHelper(node->right, temp->val);
            node->val = tempVal;
        }
    }
    // 更新节点高度
    updateHeight(node);
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node);
    // 返回子树的根节点
    return node;
}
avl_tree.kt
/* 删除节点 */
fun remove(value: Int) {
    root = removeHelper(root, value)
}

/* 递归删除节点（辅助方法） */
fun removeHelper(n: TreeNode?, value: Int): TreeNode? {
    var node = n ?: return null
    /* 1. 查找节点并删除 */
    if (value < node.value) node.left = removeHelper(node.left, value)
    else if (value > node.value) node.right = removeHelper(node.right, value)
    else {
        if (node.left == null || node.right == null) {
            val child = if (node.left != null) node.left else node.right
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child == null) return null
            else node = child
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            var temp = node.right
            while (temp!!.left != null) {
                temp = temp.left
            }
            node.right = removeHelper(node.right, temp.value)
            node.value = temp.value
        }
    }
    updateHeight(node) // 更新节点高度
    /* 2. 执行旋转操作，使该子树重新恢复平衡 */
    node = rotate(node)
    // 返回子树的根节点
    return node
}
avl_tree.rb
[class]{AVLTree}-[func]{remove}

[class]{AVLTree}-[func]{remove_helper}
avl_tree.zig
// 删除节点
fn remove(self: *Self, val: T) void {
   self.root = self.removeHelper(self.root, val).?;
}

// 递归删除节点（辅助方法）
fn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {
    var node = node_;
    if (node == null) return null;
    // 1. 查找节点并删除
    if (val < node.?.val) {
        node.?.left = self.removeHelper(node.?.left, val);
    } else if (val > node.?.val) {
        node.?.right = self.removeHelper(node.?.right, val);
    } else {
        if (node.?.left == null or node.?.right == null) {
            var child = if (node.?.left != null) node.?.left else node.?.right;
            // 子节点数量 = 0 ，直接删除 node 并返回
            if (child == null) {
                return null;
            // 子节点数量 = 1 ，直接删除 node
            } else {
                node = child;
            }
        } else {
            // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
            var temp = node.?.right;
            while (temp.?.left != null) {
                temp = temp.?.left;
            }
            node.?.right = self.removeHelper(node.?.right, temp.?.val);
            node.?.val = temp.?.val;
        }
    }
    self.updateHeight(node); // 更新节点高度
    // 2. 执行旋转操作，使该子树重新恢复平衡
    node = self.rotate(node);
    // 返回子树的根节点
    return node;
}

The node search operation in AVL trees is consistent with that in binary search trees and will not be detailed here.

7.5.4   Typical applications of AVL trees

  • Organizing and storing large amounts of data, suitable for scenarios with high-frequency searches and low-frequency intertions and removals.
  • Used to build index systems in databases.
  • Red-black trees are also a common type of balanced binary search tree. Compared to AVL trees, red-black trees have more relaxed balancing conditions, require fewer rotations for node insertion and removal, and have a higher average efficiency for node addition and removal operations.
Feel free to drop your insights, questions or suggestions