平衡二叉树

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AVL树的简介

AVL树（即平衡二叉树）是自平衡的二分查找树。在AVL树中任何节点的两个子树的高度差值为一。因此，它也是一个棵在二分查找树（BST）且为二分查找树的改进版。

注：AVL树的“AVL”来自于两位前苏联的发明者Georgy Adelson-Velsky 和 Evgenii Landis。

特点

AVL树以一棵二分查找树（BST）
自平衡：每个节点的左右子树高度相差绝对值不超过1

AVL树的实现

// avl树节点的定义typedef struct AvlNode {        int key;        struct AvlNode *left;        struct AvlNode *right;        int height;} AvlNode, *PAvlNode, *AvlTree; // avl节点生成方法PAvlNode _makeAvlNode(int key){        PAvlNode pAvlNode = (PAvlNode)malloc(sizeof(AvlNode));        pAvlNode->key = key;        pAvlNode->left = pAvlNode->right = NULL;        pAvlNode->height = 0;        return pAvlNode;}// avl树高度大小比较int Max(int h1, int h2){        return h1 > h2 ? h1 : h2;}

// avl树高度int Height(AvlTree tree) {        if (tree) {                return tree->height;        }        return 0;}// LL旋转AvlTree SingleRotateWithLeft(AvlTree tree){        AvlTree t = tree->left;        tree->left = t->right;        t->right = tree;        // 旋转之后，重新设置节点高度，即左子树或右子树高度最大值加1（本节点的高度1）        tree->height = Max(Height(tree->left), Height(tree->right)) + 1;        t->height = Max(Height(t->left), Height(t->right)) + 1;        return t;}// RR旋转AvlTree SingleRotateWithRight(AvlTree tree){        AvlTree t = tree->right;        tree->right = t->left;        t->left = tree;        tree->height = Max(Height(tree->left), Height(tree->right)) + 1;        t->height = Max(Height(t->left), Height(t->right)) + 1;        return t;}// RL旋转AvlTree DoubleRotateLeft(AvlTree tree){        tree->left = SingleRotateWithRight(tree->left);        return SingleRotateWithLeft(tree);}// LR旋转AvlTree DoubleRotateRight(AvlTree tree){        tree->right = SingleRotateWithLeft(tree->right);        return SingleRotateWithRight(tree);}// avl树节点插入方法/*    [参数]    tree : avl树根节点指针    key  : 插入avl树的关键字    [返回值]    返回关键字插入位置的指针         */AvlTree Insert(AvlTree tree, int key){        if (tree == NULL) {                tree = _makeAvlNode(key);                return tree;        } else {                if (key < tree->key) {                        tree->left = Insert(tree->left, key);                        if (Height(tree->left)                                - Height(tree->right) == 2) {                                if (key < tree->left->key)                                        tree = SingleRotateWithLeft(tree);                                else                                        tree = DoubleRotateLeft(tree);                                   }                } else if (key > tree->key) {                        tree->right = Insert(tree->right, key);                        if (Height(tree->right)                                - Height(tree->left) == 2) {                                if (key > tree->right->key)                                       tree = SingleRotateWithRight(tree);                                else                                        tree = DoubleRotateRight(tree);                        }                }        }        tree->height = Max(Height(tree->left), Height(tree->right)) + 1;        return tree;}// 单节点旋转// 需要判断节点的左右子树是否平衡，不平衡进行旋转操作AvlTree Rotate(AvlTree tree){        if (Height(tree->left) - Height(tree->right) == 2) {                if (Height(tree->left->left)                        >= Height(tree->left->right)) {                        tree = SingleRotateWithLeft(tree);                } else {                        tree = DoubleRotateLeft(tree);                }        }        if (Height(tree->right) - Height(tree->left) == 2) {                if (Height(tree->right->right)                        >= Height(tree->right->left)) {                        tree = SingleRotateWithRight(tree);                } else {                        tree = DoubleRotateRight(tree);                }        }        return tree;}// avl树节点删除AvlTree Delete(AvlTree tree, int key){        if (tree == NULL)                return NULL;        if (tree->key == key) { // 被删除节点的左右子树调整                if (tree->right == NULL) {                        AvlTree t = tree;                        tree = tree->left;                        free(t);                } else {                        AvlTree t = tree->right;                        while (t->left)                                t = t->left;                        tree->key = t->key;                        tree->right = Delete(tree->right, tree->key);                        tree->height = Max(Height(tree->left), Height(tree->right)) + 1;                }                return tree;        } else if (key > tree->key) {                tree->right = Delete(tree->right, key);        } else {                tree->left = Delete(tree->left, key);        }        // 删除节点后，重新设置其父节点的高度        tree->height = Max(Height(tree->left), Height(tree->right)) + 1;        // 删除节点后，逐级调整被删除节点的父节点        if (tree->left)                tree->left = Rotate(tree->left);        if (tree->right)                tree->right = Rotate(tree->right);        if (tree)                tree = Rotate(tree);        return tree;}// avl树遍历方法void Traversal(AvlTree tree){        if (tree) {                if (tree->left)                        Traversal(tree->left);                printf(" %d", tree->key);                if (tree->right)                        Traversal(tree->right);        }        return;}

// avl树测试方法int main(){        int a[] = {8, 2, 10, 3, 7, 1, 6};        size_t count = sizeof(a)/sizeof(a[0]);        AvlTree tree = NULL;        for (size_t i = 0; i < count; i++) {                tree = Insert(tree, a[i]);        }        Traversal(tree);        printf("\n");        Delete(tree, 7);        Traversal(tree);        printf("\n");                   return 0;}

AVL树的性能

AVL树的查找、插入和删除在平均和最坏情况下都是O(logn)。

BST、AVL和RBTree

二分查找树（BST）比较简单，但是很容易产生不平衡的问题而丧失了二分查找树的优势。
AVL树规定了左右孩子树的高度差超过1则为不平衡，为了维持平衡，在插入和删除子节点后，必须进行相应的旋转。
还有一种著名的红黑树，红黑树放宽了平衡的要求，从而减少了操作的次数。

由此AVL和RBTree都是BST的改进版。

参考

1 AVL树的旋转操作图解最详细
2 AVL树
3 平衡二叉树(AVL)实现(3)-删除
4 数据结构 – 二叉树(BST, AVLTree, RBTree, SplayTree)

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