AVL树问题

AVL树不同于平衡搜索树，它有一个平衡因子的概念。

AVL树要满足两个条件：

1、平衡搜索树（左孩子<节点<右孩子）

2、就是每个节点的左右孩子的高度查的绝对值<=1。

见代码：

/* * prototype.c * * * * Copyright (C) 2012-10-17 liutos [e-mail] */#include <stdlib.h>#include <stdio.h>#include <math.h>#include <string.h>typedef struct node {    void *value;    /* char *value; */    int height;    int bf;                     /* balance factor */    struct node *parent, *left, *right;} *AVLTree;int (*node_comparator)(struct node *node1, struct node *node2); /* Compares two nodes. If equal, returns zero; If node1 is greater then node2, returns a positive integer; otherwise, returns a negative integer. */char *(*node_name_maker)(struct node *);                        /* For generating a string represents a node. */char *(*node_value_maker)(struct node *);                       /* For generating a string represents the value of a node. */struct node *make_node(void *value){    struct node *node = malloc(sizeof(struct node));    node->value = value;    node->height = 0;    node->parent = node->left = node->right = NULL;    return node;}void free_node(struct node *node){    free(node);}int get_tree_height(AVLTree tree){    return NULL == tree? -1: tree->height;}int get_max(int a, int b){    return a > b? a: b;}int compute_tree_height(AVLTree tree){    return get_max(get_tree_height(tree->left),                   get_tree_height(tree->right)) + 1;}int compute_tree_bf(AVLTree tree){    return get_tree_height(tree->left) - get_tree_height(tree->right);}AVLTree left_rotate(AVLTree tree){    AVLTree tmp;    tmp = tree->right;    tree->right = tmp->left;    tmp->left = tree;    tmp->bf = 0;    tree->bf = 0;    tree->height = compute_tree_height(tree); /* Update the tree->height first. */    tmp->height = compute_tree_height(tmp);   /* Update the height of the root of                                                 the new tree. */    return tmp;}AVLTree right_rotate(AVLTree tree){    AVLTree tmp;    tmp = tree->left;    tree->left = tmp->right;    tmp->right = tree;    tmp->bf = 0;    tree->bf = 0;    tree->height = compute_tree_height(tree); /* The same as the left_rotate. */    tmp->height = compute_tree_height(tmp);    return tmp;}AVLTree insert_node(struct node *node, AVLTree tree){    if (NULL == tree)        return node;    if (0 == node_comparator(node, tree)) {        printf("%s already exists.\n", node_value_maker(node));        return tree;    }    if (node_comparator(node, tree) > 0) {        /* Insert into the right subtree. */        tree->right = insert_node(node, tree->right);        tree->right->parent = tree;        tree->height = compute_tree_height(tree);        tree->bf = compute_tree_bf(tree);        /* If both tree->bf and tree->right->bf are negative, it's the case of RR. */        if (-2 == tree->bf && -1 == tree->right->bf)            tree = left_rotate(tree);        /* If tree->bf is negative and tree->right->bf is positive, it's the case of RL. */        else if (-2 == tree->bf && 1 == tree->right->bf) {            tree->right = right_rotate(tree->right);            tree = left_rotate(tree);        }    } else if (node_comparator(node, tree) < 0 ) {        /* Insert into the left subtree. */        tree->left = insert_node(node, tree->left);        tree->left->parent = tree;        tree->height = compute_tree_height(tree);        tree->bf = compute_tree_bf(tree);        /* If both tree->bf and tree->left->bf are positive, it's the case of LL. */        if (2 == tree->bf && 1 == tree->left->bf)            tree = right_rotate(tree);        /* If tree->bf is positive and tree->left->bf is negative, it's the case of LR. */        else if (2 == tree->bf && -1 == tree->left->bf) {            tree->left = left_rotate(tree->left);            tree = right_rotate(tree);        }    }    return tree;}/* Free the memory of a AVLTree in postorder. */void free_tree(AVLTree tree){    struct node *node;    while (tree != NULL) {        free_tree(tree->left);        node = tree;        tree = tree->right;        free_node(node);    }}void draw_tree_core(AVLTree tree, FILE *fp){    if (NULL)        return;    if (tree->left != NULL) {        fprintf(fp, "\t%s:l -> %s;\n", node_name_maker(tree), node_name_maker(tree->left));        draw_tree_core(tree->left, fp);    }    if (tree->right != NULL) {        fprintf(fp, "\t%s:r -> %s;\n", node_name_maker(tree), node_name_maker(tree->right));        draw_tree_core(tree->right, fp);    }}void gen_node_description(struct node *node, FILE *fp){    fprintf(fp, "\t%s [label = \"<l> |{<v> value: %s| height: %d| bf: %d}|<r>\"];\n", node_name_maker(node), node_value_maker(node), node->height, node->bf);}void gen_tree_description(AVLTree tree, FILE *fp){    if (NULL == tree) return;    gen_tree_description(tree->left, fp);    gen_tree_description(tree->right, fp);    gen_node_description(tree, fp);}void draw_tree(AVLTree tree){    FILE *fp = fopen("tree.dot", "w");    if (NULL == fp) {        printf("Can not open file `tree.dot'.\n");        exit(1);    }    fprintf(fp, "digraph G {\n\tnode [shape = record];\n");    gen_tree_description(tree, fp);    draw_tree_core(tree, fp);    fprintf(fp, "}");}/* Used when the value is of type `int'. */int node_num_compare(struct node *node1, struct node *node2){    return node1->value - node2->value;}/* The string version. */int node_str_compare(struct node *node1, struct node *node2){    return strcmp(node1->value, node2->value);}/* Used when the value is of type `int'. */char *node_num_label(struct node *node){    char *label = malloc(10 * sizeof(char));    sprintf(label, "node%d", (int)node->value);    return label;}/* The string version. */char *node_str_label(struct node *node){    return (char *)node->value;}/* The string version. */char *node_str_value(struct node *node){    return (char *)node->value;}/* Used when the value is of type `int'. */char *node_num_value(struct node *node){    char *value = malloc(10 * sizeof(char));    sprintf(value, "%d", (int)node->value);    return value;}void preorder_traversal_avl(AVLTree tree){    if (NULL == tree)        return;    else {        printf("%d", (int)tree->value);        preorder_traversal_avl(tree->left);        preorder_traversal_avl(tree->right);    }}void inorder_traversal_avl(AVLTree tree){    if (NULL == tree)        return;    else {        inorder_traversal_avl(tree->left);        printf("%d", (int)tree->value);        inorder_traversal_avl(tree->right);    }}int main(int argc, char *argv[]){    /* char *datum[] = { */    /*     "ls", */    /*     "mkdir", */    /*     "rm", */    /*     "dot", */    /*     "sbcl", */    /*     "fg", */    /*     "bg", */    /*     "ghci", */    /*     "ocamlc", */    /* }; */    int datum[] = {1, 2, 3, 4, 5, 8, 7, 6, 9};    int datum_len = sizeof(datum) / sizeof(int);    int i;    AVLTree tree = NULL;    struct node *node;    node_comparator = node_num_compare;    node_name_maker = node_num_label;    node_value_maker = node_num_value;    /* node_comparator = node_str_compare; */    /* node_name_maker = node_str_label; */    /* node_value_maker = node_str_value; */    for (i = 0; i < datum_len; i++) {        node = make_node((void *)datum[i]);        tree = insert_node(node, tree);    }    /* draw_tree(tree); */    inorder_traversal_avl(tree);    printf("\n");    free_tree(tree);    return 0;}

代码来源：https://github.com/Liutos/CLab/blob/master/avl/prototype.c

实现原理：
刚开始是空树，然后每次插入一个节点（按照平衡搜索树的条件插入）。插入以后对树的节点计算平衡因子，如果不满足AVL树的要求。则进行相应的调整（所谓的旋转）。