AVL树（Adelson-Velskii-Landis tree）

AVL树是一个“加上了额外平衡条件”的二叉搜索树。其平衡条件的建立时为了确保树的深度为O(longN)。AVL树要求任何节点左右子树的高度相差不超过1。

插入操作：左-左插入和右-右插入需要单旋转；

                   左-右插入和右-左插入需要双旋转。

其实，AVL树的插入删除操作和二叉查找树的操作类似，只是需要注意调整树的平衡。

AVL树 C语言实现

删除操作的旋转示意图：

在下面的代码中，插入删除过程中，对树平衡的调整完全可以封装为函数，显得简洁，但懒人我没写成函数。

//3avl_tree.c#include "3avl_tree.h"#include "test.h"AvlTree MakeEmptyAvlTree(AvlTree avltree) {if (avltree != NULL) {MakeEmptyAvlTree(avltree->left);MakeEmptyAvlTree(avltree->right);free(avltree);}return NULL;}Position FindAvlTree(ElementType x, AvlTree avltree) {if (avltree == NULL)return NULL;if (x < avltree->element)return FindAvlTree(x, avltree->left);else if (x > avltree->element)return FindAvlTree(x, avltree->right);elsereturn avltree;}Position FindMinFromAvlTree(AvlTree avltree) {if (avltree == NULL)return NULL;else if (avltree->left == NULL)return avltree;elsereturn FindMinFromAvlTree(avltree->left);}Position FindMaxFromAvlTree(AvlTree avltree) {if (avltree == NULL)return NULL;else if (avltree->right == NULL)return avltree;elsereturn FindMaxFromAvlTree(avltree->right);}static int HeightAvlTree(AvlTree avltree) {if (avltree == NULL)return -1;elsereturn avltree->height;}static int Max(int lhs, int rhs) {return lhs > rhs ? lhs : rhs;}//左-左插入，使得k2失去平衡，因此需要在k2处单旋转static Position SingleRotateWithLeft(Position k2) {Position k1;k1 = k2->left;k2->left = k1->right;k1->right = k2;k2->height = Max(HeightAvlTree(k2->left), HeightAvlTree(k2->right)) + 1;k1->height = Max(HeightAvlTree(k1->left), HeightAvlTree(k1->right)) + 1;return k1;}//右-右插入，使得k1失去平衡，因此需要在k1处单旋转static Position SigleRotateWithRight(Position k1) {Position k2;k2 = k1->right;k1->right = k2->left;k2->left = k1;k1->height = Max(HeightAvlTree(k1->left), HeightAvlTree(k1->right)) + 1;k2->height = Max(HeightAvlTree(k2->left), HeightAvlTree(k2->right)) + 1;return k2;}//左-右插入，使得k3失去平衡，因此在k3处双旋转//双旋转其实是两次单旋转操作//先对k3的左子树进行一次右-右单旋转操作//然后对k3进行一次左-左单旋转操作static Position DoubleRotateWithLeft1(Position k3) {  //双旋转，递归k3->left = SigleRotateWithRight(k3->left);return SingleRotateWithLeft(k3);}static Position DoubleRotateWithLeft2(Position k3) {Position k1, k2;k1 = k3->left;k2 = k1->right;k1->right = k2->left;k2->left = k1;k3->left = k2->right;k2->right = k3;k1->height = Max(HeightAvlTree(k1->left), HeightAvlTree(k1->right)) + 1;k3->height = Max(HeightAvlTree(k3->left), HeightAvlTree(k3->right)) + 1;k2->height = Max(k1->height, k3->height) + 1;return k2;}//右-左插入，使得k1失去平衡，因此在k1处双旋转//双旋转其实是两次单旋转操作//先对k1的右子树进行一次左-左单旋转操作//然后对k1进行一次右-右单旋转操作static Position DoubleRotateWithRight(Position k1) {k1->right = SingleRotateWithLeft(k1->right);return SigleRotateWithRight(k1);}AvlTree InsertToAvlTree(ElementType x, AvlTree avltree) {if (avltree == NULL) { //如果树为空，分配空间，插入节点avltree = (struct AvlNode*)malloc(sizeof(struct AvlNode));if (avltree == NULL) {cout << "malloc failed." << endl;}else {avltree->element = x;avltree->left = avltree->right = NULL;}}//插入值小于根节点的值，在左子树插入节点else if (x < avltree->element) { avltree->left = InsertToAvlTree(x, avltree->left);//树失去了平衡，对树进行调整，这几个调整平衡的语句可以封装下if (HeightAvlTree(avltree->left) - HeightAvlTree(avltree->right) == 2) {if (x < avltree->left->element) //左-左，进行左单旋转avltree = SingleRotateWithLeft(avltree);else //左-右，双旋转avltree = DoubleRotateWithLeft1(avltree);}}//插入值大于根节点的值，在右子树插入节点else if (x > avltree->element) {avltree->right = InsertToAvlTree(x, avltree->right);//树失去了平衡，对树进行调整，这几个调整平衡的语句可以封装下if (HeightAvlTree(avltree->right) - HeightAvlTree(avltree->left) == 2) {if (x > avltree->right->element)avltree = SigleRotateWithRight(avltree); //右-右，单旋转else //右-左，双旋转avltree = DoubleRotateWithRight(avltree);}}avltree->height = Max(HeightAvlTree(avltree->left), HeightAvlTree(avltree->right)) + 1;return avltree;}//删除操作类似于二叉树的操作，只是需要将树调整平衡AvlTree DeleteFromAvlTree(ElementType x, AvlTree avltree) {Position temp;if (avltree == NULL)return NULL;//在左子树寻找删除位置，删除后可能导致左子树高度变小，可能要调整树if (x < avltree->element) { avltree->left = DeleteFromAvlTree(x, avltree->left);//如果右子树高度比左子树高度超过1，那么需要将树调整平衡，这几个调整平衡的语句可以封装下if (2 == (HeightAvlTree(avltree->right) - HeightAvlTree(avltree->left))) {//如果是右-左型，则要进行左单旋、右单旋；即进行如下双旋操作//否则为右-右型，需要进行左单旋操作if (HeightAvlTree(avltree->right->left) > HeightAvlTree(avltree->right->right))avltree = DoubleRotateWithRight(avltree);else avltree = SigleRotateWithRight(avltree);}}//在右子树寻找删除位置，删除后可能导致右子树高度变小，可能要调整树else if (x > avltree->element) {avltree->right = DeleteFromAvlTree(x, avltree->right);//这几个调整平衡的语句可以封装下if (2 == (HeightAvlTree(avltree->left) - HeightAvlTree(avltree->right))) {if (HeightAvlTree(avltree->left->right) > HeightAvlTree(avltree->left->left))avltree = DoubleRotateWithLeft1(avltree);elseavltree = SingleRotateWithLeft(avltree);}}//找到要删除的元素,该节点有两个孩子//那么将节点的值用右子树中最小的值A替换，然后在右子树中删除Aelse if (avltree->left && avltree->right){ avltree->element = (FindMinFromAvlTree(avltree->right))->element;avltree->right = DeleteFromAvlTree(avltree->element, avltree->right);}//找到要删除的节点，该节点为叶子节点或仅有一个孩子else {temp = avltree;if (avltree->left == NULL) //当节点只有右孩子或为叶节点时avltree = avltree->right;else if (avltree->right == NULL) //当节点只有左孩子时avltree = avltree->left;free(temp);}if (avltree)avltree->height = Max(HeightAvlTree(avltree->right), HeightAvlTree(avltree->left)) + 1;return avltree;}void VisitElement(AvlTree tree) {cout << tree->element << " ";}//先序遍历树void PreOrderTraversal(AvlTree tree) {if (tree == NULL)return;VisitElement(tree);PreOrderTraversal(tree->left);PreOrderTraversal(tree->right);}//中序遍历树void InOrderTraversal(AvlTree tree) {if (tree == NULL)return;InOrderTraversal(tree->left);VisitElement(tree);InOrderTraversal(tree->right);}int main() {AvlTree tree = NULL;tree = MakeEmptyAvlTree(tree);tree = InsertToAvlTree(1, tree);tree = InsertToAvlTree(3, tree);tree = InsertToAvlTree(5, tree);tree = InsertToAvlTree(122, tree);tree = InsertToAvlTree(7, tree);tree = InsertToAvlTree(9, tree);tree = InsertToAvlTree(11, tree);tree = InsertToAvlTree(21, tree);Position min = FindMinFromAvlTree(tree);;cout << "删除前，树高：" << HeightAvlTree(tree) << endl;cout << endl << "删除前，先序遍历：" << endl;PreOrderTraversal(tree);cout << endl << "删除前，中序遍历：" << endl;InOrderTraversal(tree);DeleteFromAvlTree(7, tree);cout << endl;cout << "删除后，树高：" << HeightAvlTree(tree) << endl;cout << endl << "删除后，先序遍历：" << endl;PreOrderTraversal(tree);cout << endl << "删除后，中序遍历：" << endl;InOrderTraversal(tree);}

//3avl_tree.htypedef int ElementType;#ifndef TEST_AVL_TREE_H#define TEST_AVL_TREE_Hstruct AvlNode;typedef struct AvlNode *Position;typedef struct AvlNode *AvlTree;struct AvlNode {ElementType element;AvlTree left;AvlTree right;int height;};AvlTree MakeEmptyAvlTree(AvlTree avltree);Position FindAvlTree(ElementType x, AvlTree avltree);Position FindMinFromAvlTree(AvlTree avltree);Position FindMaxFromAvlTree(AvlTree avltree);AvlTree InsertToAvlTree(ElementType x, AvlTree avltree);AvlTree DeleteFromAvlTree(ElementType x, AvlTree avltree);ElementType RetrieveAvlTree(Position pos);#endif

程序执行结果：

Avl的C++实现