AVL树详解

AVL树定义：

AVL树又称为高度平衡的二叉搜索树。它能保持二叉树的高度平衡，尽量降低二叉树的高度，减少树的平均搜索长度。

AVL树性质：

1. 左子树和右子树的高度之差的绝对值不超过1
2. 树中的每个左子树和右子树都是AVL树
3. 每个节点都有一个平衡因子(balance factor--bf),任一节点的平衡因子是-1,0,1。(每个节点的平衡因子等于右子树的高度减去左子树的高度 )   

AVL树效率：

一棵AVL树有N个节点，其高度可以保持在lgN，插入/删除/查找的时间复杂度也是lgN。

AVL树节点信息如下：

template<class K, class V>struct AVLTreeNode{K _key;  //关键字V _value; AVLTreeNode<K, V>* _parent;//父亲AVLTreeNode<K, V>* _left;//左孩子AVLTreeNode<K, V>* _right;//右孩子int _bf;//平衡因子AVLTreeNode<K, V>(const K& key, const V& value)//构造函数: _key(key), _value(value), _parent(NULL), _left(NULL), _right(NULL), _bf(0){}};

AVL树操作包括：插入，查找，删除等。

由于AVL树要求左右子树高度差的绝对值不超过1，因此AVL树必须通过旋转调平衡因子保持平衡。

插入：先插入节点再调平衡因子。

1、插入节点：

（1）若AVL树为空则new一个节点作为根节点；

        （2）若要插入的节点等于根节点，返回false；

        （3）若要插入的节点大于根节点，递归右子树插入节点，反之递归左子树插入节点。

2、调平衡因子（通过旋转）

void _RotateL(Node*& parent)  // '\'  ->   '/\'{Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL){subRL->_parent = parent;}subR->_left = parent;subR->_parent = parent->_parent;Node* ppNode = parent->_parent;parent->_parent = subR;//更新节点信息if (ppNode == NULL){_root = subR;subR->_parent = NULL;}else if (ppNode->_left == parent){ppNode->_left = subR;}else{ppNode->_right = subR;}subR->_parent = ppNode;parent->_bf = subR->_bf = 0;//更新平衡因子}

void _RotateR(Node*& parent)   // '/'  ->   '/\'{Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR){subLR->_parent = parent;}subL->_right = parent;subL->_parent = parent->_parent;Node* ppNode = parent->_parent;parent->_parent = subL;//更新节点信息if (ppNode == NULL){_root = subL;subL->_parent = NULL;}else if (ppNode->_left == parent){ppNode->_left = subL;}else{ppNode->_right = subL;}subL->_parent = ppNode;parent->_bf = subL->_bf = 0;//更新平衡因子}

查找：

1、若搜索树为空则返回空；

        2、若要查找的节点等于根节点，返回根节点；

        3、若要查找的节点大于根节点，递归查找右子树，反之递归查找左子树。

Node* Find(const K& key){if (_root == NULL){return NULL;}if (key > _root->_key){_root = _root->_right;}else if (key < _root->_key){_root = _root->_left;}else{return _root;}return NULL;}

此外还有删除操作，先删除节点再调平衡因子。删除方法和搜索二叉树删除方法（http://blog.csdn.net/wanglelelihuanhuan/article/details/51585941）一样

完整代码如下：

AVLTree.h

#include<math.h>template<class K, class V>struct AVLTreeNode{K _key;  //关键字V _value; AVLTreeNode<K, V>* _parent;//父亲AVLTreeNode<K, V>* _left;//左孩子AVLTreeNode<K, V>* _right;//右孩子int _bf;//平衡因子AVLTreeNode<K, V>(const K& key, const V& value)//构造函数: _key(key), _value(value), _parent(NULL), _left(NULL), _right(NULL), _bf(0){}};template<class K, class V>class AVLTree{typedef AVLTreeNode<K, V> Node;public:AVLTree():_root(NULL){}Node* Find(const K& key){if (_root == NULL){return NULL;}if (key > _root->_key){_root = _root->_right;}else if (key < _root->_key){_root = _root->_left;}else{return _root;}return NULL;}bool Insert(const K& key,const V& value){//先插入节点if (_root == NULL){_root = new Node(key, value);return true;}Node* parent = NULL;Node* cur = _root;while (cur){if (key > cur->_key){parent = cur;cur = cur->_right;}else if (key < cur->_key){parent = cur;cur = cur->_left;}else{return false;}}cur = new Node(key, value);if (key > parent->_key){parent->_right = cur;cur->_parent = parent;}else{parent->_left = cur;cur->_parent = parent;}//再调平衡因子while (parent){if (parent->_left == cur){parent->_bf--;}else{parent->_bf++;}if (parent->_bf == 0)  //已经是平衡树{break;}else if (parent->_bf == -1 || parent->_bf == 1) //继续回溯{cur = parent;parent = cur->_parent;}else // -2  2  调整 旋转{if (parent->_bf == 2){if (cur->_bf == 1)  //  '\'   左单旋  {_RotateL(parent);}else  //-1   '>' 右左双旋{_RotateRL(parent);}}else  //-2{if (cur->_bf == -1)  // '/' 右单旋{_RotateR(parent);}else   // '<' 左右双旋{_RotateLR(parent);}}break;}}return true;}void Inorder(){_Inorder(_root);cout << endl;}bool IsBalance(){return _IsBalance(_root);}protected:void _RotateL(Node*& parent)  // '\'  ->   '/\'{Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL){subRL->_parent = parent;}subR->_left = parent;subR->_parent = parent->_parent;Node* ppNode = parent->_parent;parent->_parent = subR;//更新节点信息if (ppNode == NULL){_root = subR;subR->_parent = NULL;}else if (ppNode->_left == parent){ppNode->_left = subR;}else{ppNode->_right = subR;}subR->_parent = ppNode;parent->_bf = subR->_bf = 0;//更新平衡因子}void _RotateR(Node*& parent)   // '/'  ->   '/\'{Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR){subLR->_parent = parent;}subL->_right = parent;subL->_parent = parent->_parent;Node* ppNode = parent->_parent;parent->_parent = subL;//更新节点信息if (ppNode == NULL){_root = subL;subL->_parent = NULL;}else if (ppNode->_left == parent){ppNode->_left = subL;}else{ppNode->_right = subL;}subL->_parent = ppNode;parent->_bf = subL->_bf = 0;//更新平衡因子}void _RotateLR(Node*& parent)   //  '<'  ->   '/'  ->   '/\'{Node* subL = parent->_left;Node* subLR = subL->_right;int bf = subLR->_bf;_RotateL(parent->_left);_RotateR(parent);if (bf == 1){subL->_bf = -1;parent->_bf = 0;}else if (bf == -1){subL->_bf = 0;parent->_bf = 1;}else{subL->_bf = parent->_bf = 0;}subLR->_bf = 0;}void _RotateRL(Node*& parent){Node* subR = parent->_right;Node* subRL = subR->_left;int bf = subRL->_bf;_RotateR(parent->_right);_RotateL(parent);if (bf == 1){subR->_bf = 0;parent->_bf = -1;}else if (bf == -1){subR->_bf = 1;parent->_bf = 0;}else{subR->_bf = parent->_bf = 0;}subRL->_bf = 0;}int _Height(Node* root){if (root == NULL){return 0;}int Left = _Height(root->_left);int Right = _Height(root->_right);return (Left > Right ?  Left:Right)+1;}void _Inorder(Node* root){if (root == NULL)return ;_Inorder(root->_left);cout << root->_key << " ";_Inorder(root->_right);}bool _IsBalance(Node*& root){if (root == NULL){return true;}int bf = _Height(root->_right) - _Height(root->_left) ;if (abs(bf) > 1 || bf != root->_bf){cout << "平衡因子有问题：" << root->_key << endl;return false;}return _IsBalance(root->_left) && _IsBalance(root->_right);}protected:Node* _root;};void TestInsert(){int arr1[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };AVLTree<int, int> alt1;for (int i = 0; i < sizeof(arr1) / sizeof(arr1[0]); ++i){alt1.Insert(arr1[i],i);}alt1.Inorder();cout << "isBlance? " << alt1.IsBalance() << endl;int arr[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16,14};AVLTree<int, int> alt2;for (int i = 0; i < sizeof(arr) / sizeof(arr[0]); ++i){alt2.Insert(arr[i], i);}alt2.Inorder();cout << "isBlance? " << alt2.IsBalance() << endl;}

Test.cpp

#include<iostream>using namespace std;#include"AVLTree.h"int main(){TestInsert();return 0;}