数据结构】AVL树

AVL树，是一棵平衡搜索二叉树，既满足搜索树的性质（见二叉搜索树的文章，链接：二叉搜索树），又满足平衡树

的性质（左右子树的高度差不大于2）。

在二叉搜索树中，我们知道要插入一个元素，必须将他插到合适的位置，但是在AVL树中，不仅要插入到合适的位

置，还要保证插入该元素之后这棵树是平衡搜索二叉树。

关于如何调整一棵二叉树为平衡二叉树，这里就涉及到四种旋转：

左单旋，右单旋，左右双旋，右左双旋。

下边就每一种旋转都给出图示：

左单旋：

右单旋：与左单旋同理。

右左双旋：

左右双旋：情形类似于右左双旋。

说到平衡树，那么如何判断一棵二叉树是不是平衡二叉树？？？

思路一：如果我们可以计算出左子树的高度和右子树的高度，两者做差看是否符合平衡树的条件。这样一来，我们会

遍历这棵树两次，导致效率就能低下一些。

思路二：如果我们每一次遍历，都会带回树的高度，这样就会少遍历一次，时间复杂度是0（N）。下边给出完整代

码：

#pragma once#include<iostream>using namespace std;#include<stack>template<typename K,typename V>struct AVLTreeNode{K _key;V _value;AVLTreeNode<K, V>* _left;AVLTreeNode<K, V>* _right;AVLTreeNode<K, V>* _parent;int _bf;AVLTreeNode(const K& key,const V& value = 0):_key(key),_value(value),_left(NULL),_right(NULL),_parent(NULL),_bf(0){}};template<typename K, typename V>class AVLTree{typedef AVLTreeNode<K, V> Node;public:AVLTree():_root(NULL){}~AVLTree(){_Destroy(_root);}bool InsertNonR(const K& key){if (_root == NULL){_root = new Node(key);_root->_parent = NULL;return true;}Node* cur = _root;Node* parent = _root;while (cur){//找插入位置if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}//无法插入elsereturn false;} cur = new Node(key); cur->_parent = parent;if (parent->_key < key){parent->_right = cur;}else{parent->_left = cur;}while (parent)//调整到根节点之后就不再调整{//if (parent->_parent && parent->_parent->_left == parent)if(cur == parent->_left)--parent->_bf;else if (cur == parent->_right)             //if (parent->_parent->_right == parent)++parent->_bf;if (parent->_bf == 0)//说明parent的高度并没有改变，此时调整结束return true;else if (parent->_bf == 1 || parent->_bf == -1){cur = parent;parent = parent->_parent;}else{if (parent->_bf == 2)//树已经不平衡，需要调整{if (cur->_bf == 1)_RotateL(parent);else             //if (cur->_bf == -1)_RotateRL(parent);}//if (parent->_bf == -2)else{if (cur->_bf == -1)_RotateR(parent);else                 //if (cur->_bf == 1)_RotateLR(parent);}}}return true;}void InOrderNonR(){if (_root == NULL)return;stack<Node*> s;Node* cur = _root;while (cur || !s.empty()){while (cur){s.push(cur);cur = cur->_left;}Node* top = s.top();s.pop();cout << top->_key << " ";cur = top->_right;}cout << endl;}bool IsBalance(){return _IsBalance(_root);}bool IsBalanceOP(){int height = 0;return _IsBalanceOP(_root,height);}size_t Height(){return _Height(_root);}bool Remove(const K& key){if (_root == NULL)return false;Node* cur = _root;Node* parent = NULL;Node* del = NULL;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else//找到所要删除的元素{if (cur->_left == NULL && cur->_right == NULL)//叶子结点{if (parent && parent->_left == cur){parent->_left = NULL;parent->_bf++;del = cur;}else if (parent && parent->_right == cur){parent->_right = NULL;parent->_bf--;del = cur;}else{del = _root;_root->_parent = NULL;}}else if (cur->_left == NULL || cur->_right == NULL){if (cur->_right != NULL){if (parent == NULL){_root = cur->_right;_root->_parent = NULL;}if (parent->_right == cur)//只有右孩子{parent->_right = cur->_right;parent->_bf--;}else if (parent->_left == cur){parent->_left = cur->_right;++parent->_bf;}}else//只有左孩子{if (parent == NULL){_root = cur->_left;_root->_parent = NULL;}else if (parent->_left == cur){parent->_left = cur->_left;parent->_bf++;}else{parent->_right = cur->_left;parent->_bf--;}}del = cur;}else//有左右孩子{Node* minRight = cur->_right;//找右子树的最左结点while (minRight->_left){parent = minRight;minRight = minRight->_left;}cur->_key = minRight->_key;if (parent->_left == minRight){parent->_left = minRight->_right;parent->_bf++;}else if (parent->_right == minRight){parent->_right = minRight->_right;parent->_bf--;}del = minRight;}while (parent){cur = del;if (cur == parent->_left)++parent->_bf;else if (cur == parent->_right)--parent->_bf;if (parent->_bf == 0){cur = parent;parent = parent->_parent;}else if (parent->_bf == 1 || parent->_bf == -1){//高度没有改变，直接跳出break;}else{if (parent->_bf == 2)//树已经不平衡，需要调整{if (cur->_bf == 1)_RotateL(parent);else             //if (cur->_bf == -1)_RotateRL(parent);}//if (parent->_bf == -2)else{if (cur->_bf == -1)_RotateR(parent);else                 //if (cur->_bf == 1)_RotateLR(parent);}}}delete del;del = NULL;return true;}}return false;}protected://优化版本bool _IsBalanceOP(Node* root,int& height){if (root == NULL){height = 0;return true;}int leftHeight = 0;if (!_IsBalanceOP(root->_left, leftHeight))return false;int rightHeight = 0;if (!_IsBalanceOP(root->_right, rightHeight))return false;height = 1 + rightHeight > leftHeight ? rightHeight : leftHeight;return true;}size_t _Height(Node* root){if (root == NULL)return 0;int leftHeight = _Height(root->_left);int rightHeight = _Height(root->_right);return leftHeight > rightHeight ? leftHeight + 1 : rightHeight + 1;}//大多数结点会计算两次，时间复杂度0(N*N)bool _IsBalance(Node* root){if (root == NULL)return true;int leftHeight = _Height(root->_left);int rightHeight = _Height(root->_right);if (rightHeight - leftHeight != root->_bf){cout << "平衡因子异常" << root->_key << " ";}return abs(rightHeight - leftHeight) < 2&& _IsBalance(root->_left)&& _IsBalance(root->_right);}void _RotateR(Node* parent){Node* subL = parent->_left;Node* ppNode = parent->_parent;//记住parent的父节点Node*  subLR = subL->_right;//parent与subLR连接parent->_left = subLR;if (subLR != NULL)subLR->_parent = parent;//parent与subL连接subL->_right = parent;parent->_parent = subL;//ppNode与subL进行连接if (ppNode == NULL){_root = subL;subL->_parent = NULL;}else{if (ppNode->_left == parent)ppNode->_left = subL;elseppNode->_right = subL;subL->_parent = ppNode;}subL->_bf = parent->_bf = 0;}void _RotateL(Node* parent){Node* subR = parent->_right;Node* ppNode = parent->_parent;//记住parent的父节点Node* subRL = subR->_left;//subRL与parent相连parent->_right = subRL;if (subRL != NULL)subRL->_parent = parent;//subR与parent相连接subR->_left = parent;parent->_parent = subR;//ppNode与subR连接if (ppNode == NULL){_root = subR;subR->_parent = NULL;}else{if (ppNode->_left == parent)ppNode->_left = subR;elseppNode->_right = subR;subR->_parent = ppNode;}parent->_bf = subR->_bf = 0;}void _RotateLR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;int bf = subLR->_bf;_RotateL(parent->_left);_RotateR(parent);if (bf == 0){parent->_bf = 0;subL->_bf = 0;subLR->_bf = 0;}else if (bf == 1){parent->_bf = 0;subL->_bf = -1;subLR->_bf = 1;}else // bf == -1{parent->_bf = 1;subL->_bf = 0;subLR->_bf = -1;}}void _RotateRL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;int bf = subRL->_bf;_RotateR(parent->_right);_RotateL(parent);if (bf == 0){parent->_bf = 0;subR->_bf = 0;}else if (bf == 1){parent->_bf = -1;subR->_bf = 0;subRL->_bf = 1;}else{parent->_bf = 0;subR->_bf = 1;subRL->_bf = -1;}}void _Destroy(Node* root){if (root == NULL)return;_Destroy(root->_left);_Destroy(root->_right);delete root;}private:Node* _root;};void TestAVL(){AVLTree<int, int> tree1;int array1[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };for (int i = 0; i < sizeof(array1) / sizeof(array1[0]); ++i){tree1.InsertNonR(array1[i]);}tree1.InOrderNonR();cout <<"IsBalance?"<< tree1.IsBalance() << endl;cout << "IsBalance?" << tree1.IsBalanceOP() << endl;tree1.Remove(16);tree1.Remove(3);tree1.Remove(7);tree1.Remove(11);tree1.InOrderNonR();//9 26 18 14 15tree1.Remove(9);tree1.Remove(26);tree1.Remove(18);tree1.Remove(15);tree1.InOrderNonR();//9 26 18 14 15cout << "IsBalance?" << tree1.IsBalance() << endl;int array2[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 };AVLTree<int, int> tree2;for (size_t i = 0;i<sizeof(array2) / sizeof(array2[0]);++i){tree2.InsertNonR(array2[i]);}tree2.InOrderNonR();cout << "IsBalance?" << tree2.IsBalance() << endl;cout << "IsBalance?" << tree2.IsBalanceOP() << endl;tree2.Remove(5);tree2.InOrderNonR();}