AVL树(平衡二叉树)

AVL树是排序二叉树的优化版，多了个调整操作，排序二叉树在某些情况下可能会变的跟链表差不多，比如连续插入多个非递减的数：

这就会使得各种操作非常费时间，几乎和链表一样。但是AVL树避免了这一点，保证了任意一个节点的左右儿子节点的高度差不会超过1，这就使得AVL树的复杂度很平衡(插入删除查找log(n))。
这里的调整操作是基于两个旋转操作来进行的，即左旋和右旋：
右旋：

左旋(实际上就是右旋的反转)：

当出现某个节点的左右儿子的高度差距太大时，就要进行调整使左右平衡。当出现类似 这种左儿子高度比右儿子高度大2且左儿子的左儿子比左儿子的右儿子高度大1的时候这种暂且叫做LL，然后我们需要一个右旋操作来进行平衡，同理RR(其实是因为作图实在太丑了就不想做了.)就是左儿子高度比右儿子高度小2且左儿子的左儿子比左儿子的右儿子高度小1。然后就是LR跟RL（两种情况一样）：

从图就可以看出来，这种情况需要进行先左旋然后右旋两个旋转操作，RL就是先右旋然后左旋。经过这样的平衡操作之后二叉树就会处于一个比较平衡的状态。

首先是节点的定义：

struct node {    node *left, *right;    int val;   // 值    int h;  // 节点高度};

必要函数：

int height(node *rt) {  // 获取某个节点的高度    if (!rt) return 0;    return rt->h;}node *getNewNode(int val) {  // 新建节点    node *rt = new node;    rt->left = rt->right = NULL;    rt->h = 1;    rt->val = val;    return rt;}node *singleLeftRotate(node *a) {  // 左旋操作    node *b = a->right;    a->right = b->left;    b->left = a;    a->h = max(height(a->left), height(a->right)) + 1;    b->h = max(height(b->left), height(b->right)) + 1;    return b;}node *singleRightRotate(node *a) {  // 右旋操作    node *b = a->left;    a->left = b->right;    b->right = a;    a->h = max(height(a->left), height(a->right)) + 1;    b->h = max(height(b->left), height(b->right)) + 1;    return b;}void adjust(node* &rt) {  // 调整函数，经过的每个节点都要进行检查    rt->h = max(height(rt->left), height(rt->right)) + 1;    if (height(rt->left) - height(rt->right) == 2) {        if (height(rt->left->left) - height(rt->left->right) == 1) { // LL            rt = singleRightRotate(rt);        } else { // LR            rt->left = singleLeftRotate(rt->left);            rt = singleRightRotate(rt);        }    } else if (height(rt->right) - height(rt->left) == 2) {        if (height(rt->right->right) - height(rt->right->left) == 1) {  // RR            rt = singleLeftRotate(rt);        } else { // RL            rt->right = singleRightRotate(rt->right);            rt = singleLeftRotate(rt);        }    }    rt->h = max(height(rt->left), height(rt->right)) + 1;}

以上是AVL树的核心函数，然后就是插入删除操作：

node *Insert(node* &rt, int val) {    if (!rt) rt = getNewNode(val);    else if (val > rt->val) rt->right = Insert(rt->right, val);    else if (val < rt->val) rt->left = Insert(rt->left, val);    else { }    adjust(rt);    return rt;}node *deleteNode(node *rt, int val) {  // 删除操作稍微麻烦一点，原理就是一旦找到要删除的值，将这个数与比它小的数中最小的(实际上就是从这个节点的左节点往下，每次都走左节点直到不能走 最大一个同理)或者比它大的数中最大的互换位置，然后这个数就到了叶子节点上去了，然后就直接删除就行了    if (val > rt->val) rt->right = deleteNode(rt->right, val);    else if (val < rt->val) rt->left = deleteNode(rt->left, val);    else {        if (rt->left) {            node* dn = NULL;            for (dn = rt->left; NULL != dn->right; dn = dn->right);            rt->val = dn->val;            rt->left = deleteNode(rt->left, dn->val);        } else if (NULL != rt->right) {            node* dn = NULL;            for (dn = rt->right; NULL != dn->left; dn = dn->left);            rt->val = dn->val;            rt->right = deleteNode(rt->right, dn->val);        } else {            free(rt);            return NULL;        }    }    adjust(rt);    return rt;}

我觉得代码非常好理解的。。

#include <iostream>#include <algorithm>#include <cstdio>#include <cstring>#include <cstdlib>using namespace std;struct node {    node *left, *right;    int val;    int h;};int height(node *rt) {    if (!rt) return 0;    return rt->h;}node *getNewNode(int val) {    node *rt = new node;    rt->left = rt->right = NULL;    rt->h = 1;    rt->val = val;    return rt;}node *singleLeftRotate(node *a) {    node *b = a->right;    a->right = b->left;    b->left = a;    a->h = max(height(a->left), height(a->right)) + 1;    b->h = max(height(b->left), height(b->right)) + 1;    return b;}node *singleRightRotate(node *a) {    node *b = a->left;    a->left = b->right;    b->right = a;    a->h = max(height(a->left), height(a->right)) + 1;    b->h = max(height(b->left), height(b->right)) + 1;    return b;}struct node *root;void adjust(node* &rt) {    rt->h = max(height(rt->left), height(rt->right)) + 1;    if (height(rt->left) - height(rt->right) == 2) {        if (height(rt->left->left) - height(rt->left->right) == 1) {            rt = singleRightRotate(rt);        } else {            rt->left = singleLeftRotate(rt->left);            rt = singleRightRotate(rt);        }    } else if (height(rt->right) - height(rt->left) == 2) {        if (height(rt->right->right) - height(rt->right->left) == 1) {            rt = singleLeftRotate(rt);        } else {            rt->right = singleRightRotate(rt->right);            rt = singleLeftRotate(rt);        }    }    rt->h = max(height(rt->left), height(rt->right)) + 1;}node *Insert(node* &rt, int val) {    if (!rt) rt = getNewNode(val);    else if (val > rt->val) rt->right = Insert(rt->right, val);    else if (val < rt->val) rt->left = Insert(rt->left, val);    else { }    adjust(rt);    return rt;}node *deleteNode(node *rt, int val) {    if (val > rt->val) rt->right = deleteNode(rt->right, val);    else if (val < rt->val) rt->left = deleteNode(rt->left, val);    else {        if (rt->left) {            node* dn = NULL;            for (dn = rt->left; NULL != dn->right; dn = dn->right);            rt->val = dn->val;            rt->left = deleteNode(rt->left, dn->val);        } else if (NULL != rt->right) {            node* dn = NULL;            for (dn = rt->right; NULL != dn->left; dn = dn->left);            rt->val = dn->val;            rt->right = deleteNode(rt->right, dn->val);        } else {            free(rt);            return NULL;        }    }    adjust(rt);    return rt;}void dfs(node *rt) {    if (rt->left) dfs(rt->left);    printf("%d ", rt->val);    if (rt->right) dfs(rt->right);}int m, n, val;int main() {//    freopen("out.txt", "r", stdin);    scanf("%d", &n);    for (int i = 0; i < n; ++i) {        scanf("%d", &val);        if (i == 0) root = getNewNode(val);        else root = Insert(root, val);    }    dfs(root);    printf("\n");    scanf("%d", &m);    for (int i = 0; i < m; ++i) {        scanf("%d", &val);        root = deleteNode(root, val);    }    dfs(root);    printf("\n");    return 0;}