AVL Tree 平衡二叉树基本插入删除节点功能的实现

简述：

实现AVL 树，主要是两个功能 ： 插入某节点和删除某节点

AVL Tree的定义，

1. 是一棵二叉搜索树（故而每个节点是惟一的, 如果出现重复数字会破坏平衡树的算法）

2. 每个节点左右子树的高度之差（平衡因子）相差最多为1

实现：

为了使所得的二叉树为平衡二叉树，

首先在BSTNode中加了一个计算节点高度的方法getHeight()， 当两个节点高度相差2的时候，视为平衡破坏

int getHeight(){if(this == NULL)return 0;if(left == NULL && right == NULL)return 1;else{return 1 + max(left->getHeight(), right->getHeight());}}

之后讨论一下，不平衡出现的四种情况, 新增节点（红色）

1） LL，  新建在左子树的左节点上

LL代码实现：

template<class Type>BSTNode<Type>* AVLTree<Type>::LL(BSTNode<Type>* &topNode){BSTNode<Type> * leftSonNode = topNode->left;topNode->left = leftSonNode->right;leftSonNode->right = topNode;return leftSonNode;}

2） RR，  新建在右子树的右节点上

RR代码实现：

template<class Type>BSTNode<Type>* AVLTree<Type>::RR(BSTNode<Type>* &topNode){BSTNode<Type> *rightSonNode = topNode->right;topNode->right = rightSonNode->left;rightSonNode->left = topNode;return rightSonNode;}

3） LR，  新建在左子树的右节点上

LR代码实现：

template<class Type>BSTNode<Type>* AVLTree<Type>::LR(BSTNode<Type>* &topNode){topNode->left = RR(topNode->left);return LL(topNode);}

4） RL，  新建在右子树的左节点上

RL代码实现：

template<class Type>BSTNode<Type>* AVLTree<Type>::RL(BSTNode<Type>* &topNode){topNode->right = LL(topNode->right);return RR(topNode);}

对于删除操作，每一次删除一个节点之后，优先考虑其子节点的最左节点的值替换删除节点，但是需要注意的是，

在替换之后需要自修改节点向下每个节点做Rotate操作，用来处理因为删除某个节点之后平衡树的破坏

下面是Delete函数的实现：

template<class Type>BSTNode<Type>* AVLTree<Type>::Delete(const Type& key){return root = Delete(root, key);}template<class Type>BSTNode<Type>* AVLTree<Type>::Delete(BSTNode<Type>* &node, const Type &key){if(node == NULL){return NULL;}/** * if we find the matched key, * delete the matched node and replace it by the most left node * of its right child */else if(key == node->key){if(!node->right){BSTNode<Type> *newNode = node->left;delete node;return newNode;}else{BSTNode<Type> *secondMostLeftNode = node->right;if(secondMostLeftNode->left == NULL){return secondMostLeftNode;}while(secondMostLeftNode->left->left)secondMostLeftNode = secondMostLeftNode->left;BSTNode<Type> *mostLeftNode = secondMostLeftNode->left;secondMostLeftNode->left->left = node->left;secondMostLeftNode->left->right = node->right;secondMostLeftNode->left = NULL;return mostLeftNode;}}//from bottom to the topelse if(key < node->key){node->left = Delete(node->left, key);}else{node->right = Delete(node->right, key);}if(node->left)node->left = Rotate(node->left);if(node->right)node->right = Rotate(node->right);node = Rotate(node);return node;}/** * Rotate one node and its sub tree */template<class Type>BSTNode<Type>* AVLTree<Type>::Rotate(BSTNode<Type>* node){if(node->left->getHeight() - node->right->getHeight() == 2){if(node->left->left->getHeight() >= node->left->right->getHeight())node = LL(node);elsenode = LR(node);}if(node->right->getHeight() - node->left->getHeight() == 2){if(node->right->right->getHeight() >= node->right->left->getHeight())node = RR(node);elsenode = RL(node);}return node;}

之后是整个平衡树插入删除节点实现以及在main函数中测试的代码：

#include <iostream>#include <ctime>#include <cstdlib>using namespace std;template<class Type>class AVLTree;/** * Binary Search Tree Node:  BSTNode class */template<class Type>class BSTNode{friend class AVLTree<Type>;private:Type key;BSTNode *left;BSTNode *right;public:BSTNode(): left(NULL), right(NULL){}BSTNode(const Type& key): key(key), left(NULL), right(NULL){}Type getkey(){return key;}int getHeight(){if(this == NULL)return 0;if(left == NULL && right == NULL)return 1;else{return 1 + max(left->getHeight(), right->getHeight());}}void clear(){if(this == NULL)return;left->clear();right->clear();delete this;}void Output_DLR(){ //Node -> left -> Right orderif(this != NULL){cout << key << ", ";left->Output_DLR();right->Output_DLR();}}};/** * AVLTree class */template<class Type>class AVLTree{private:BSTNode<Type> *root;public:AVLTree(): root(NULL){}BSTNode<Type>* Insert(BSTNode<Type>* &, const Type&);BSTNode<Type>* Insert(const Type& );BSTNode<Type>* Delete(BSTNode<Type>* &, const Type&);BSTNode<Type>* Delete(const Type& );BSTNode<Type>* Rotate(BSTNode<Type>* );BSTNode<Type>* GetRoot();BSTNode<Type>* LL(BSTNode<Type>* &);BSTNode<Type>* LR(BSTNode<Type>* &);BSTNode<Type>* RL(BSTNode<Type>* &);BSTNode<Type>* RR(BSTNode<Type>* &);void Clear();void Output_DLR();void Output_LRN();};template<class Type>BSTNode<Type>* AVLTree<Type>::LL(BSTNode<Type>* &topNode){BSTNode<Type> * leftSonNode = topNode->left;topNode->left = leftSonNode->right;leftSonNode->right = topNode;return leftSonNode;}template<class Type>BSTNode<Type>* AVLTree<Type>::RR(BSTNode<Type>* &topNode){BSTNode<Type> *rightSonNode = topNode->right;topNode->right = rightSonNode->left;rightSonNode->left = topNode;return rightSonNode;}template<class Type>BSTNode<Type>* AVLTree<Type>::LR(BSTNode<Type>* &topNode){topNode->left = RR(topNode->left);return LL(topNode);}template<class Type>BSTNode<Type>* AVLTree<Type>::RL(BSTNode<Type>* &topNode){topNode->right = LL(topNode->right);return RR(topNode);}template<class Type>BSTNode<Type>* AVLTree<Type>::GetRoot(){return root;}template<class Type>BSTNode<Type>* AVLTree<Type>::Insert(const Type& key){return Insert(root, key);}template<class Type>BSTNode<Type>* AVLTree<Type>::Insert(BSTNode<Type>* &node, const Type &key){if(node == NULL){return node = new BSTNode<Type>(key);}//from bottom to the topelse if(key < node->key){Insert(node->left, key);if(node->left->getHeight() - node->right->getHeight() == 2){if(key < node->left->key)node = LL(node);elsenode = LR(node);}}else{Insert(node->right, key);if(node->right->getHeight() - node->left->getHeight() == 2){if(key > node->right->key)node = RR(node);elsenode = RL(node);}}return node;}template<class Type>BSTNode<Type>* AVLTree<Type>::Delete(const Type& key){return root = Delete(root, key);}template<class Type>BSTNode<Type>* AVLTree<Type>::Delete(BSTNode<Type>* &node, const Type &key){if(node == NULL){return NULL;}/** * if we find the matched key, * delete the matched node and replace it by the most left node * of its right child */else if(key == node->key){if(!node->right){BSTNode<Type> *newNode = node->left;delete node;return newNode;}else{BSTNode<Type> *secondMostLeftNode = node->right;if(secondMostLeftNode->left == NULL){return secondMostLeftNode;}while(secondMostLeftNode->left->left)secondMostLeftNode = secondMostLeftNode->left;BSTNode<Type> *mostLeftNode = secondMostLeftNode->left;secondMostLeftNode->left->left = node->left;secondMostLeftNode->left->right = node->right;secondMostLeftNode->left = NULL;return mostLeftNode;}}//from bottom to the topelse if(key < node->key){node->left = Delete(node->left, key);}else{node->right = Delete(node->right, key);}if(node->left)node->left = Rotate(node->left);if(node->right)node->right = Rotate(node->right);node = Rotate(node);return node;}/** * Rotate one node and its sub tree */template<class Type>BSTNode<Type>* AVLTree<Type>::Rotate(BSTNode<Type>* node){if(node->left->getHeight() - node->right->getHeight() == 2){if(node->left->left->getHeight() >= node->left->right->getHeight())node = LL(node);elsenode = LR(node);}if(node->right->getHeight() - node->left->getHeight() == 2){if(node->right->right->getHeight() >= node->right->left->getHeight())node = RR(node);elsenode = RL(node);}return node;}template<class Type>void AVLTree<Type>::Clear(){root->clear();root = NULL;}template<class Type>void AVLTree<Type>::Output_DLR(){if(!root)cout << "EMPTY TREE! " << endl;elseroot->Output_DLR();}template<class Type>void AVLTree<Type>::Output_LRN(){if(!root)cout << "EMPTY TREE! " << endl;elseroot->Output_LRN();}//Test Mainint main() {AVLTree<int> *tree = new AVLTree<int>();cout << "First, Test Insert(key) funciton: " << endl;    cout << "Test LL : " << endl;    //test LL    tree->Insert(8);    tree->Insert(6);    tree->Insert(11);    tree->Insert(4);    tree->Insert(7);    tree->Insert(2);    cout << "DLR Output LL: " << endl;    tree->GetRoot()->Output_DLR();    tree->Clear();    //test RR    cout << endl << endl << "Test RR : " << endl;    tree->Insert(8);    tree->Insert(6);    tree->Insert(10);    tree->Insert(9);    tree->Insert(12);    tree->Insert(14);    cout << "DLR Output RR: " << endl;    tree->GetRoot()->Output_DLR();    tree->Clear();    //test LR    cout << endl << endl << "Test LR : " << endl;    tree->Insert(9);    tree->Insert(6);    tree->Insert(11);    tree->Insert(4);    tree->Insert(7);    tree->Insert(8);    cout << "DLR Output LR: " << endl;    tree->GetRoot()->Output_DLR();    tree->Clear();    //test RL    cout << endl << endl << "Test RL : " << endl;    tree->Insert(6);    tree->Insert(4);    tree->Insert(12);    tree->Insert(10);    tree->Insert(14);    tree->Insert(8);    cout << "DLR Output RL: " << endl;    tree->GetRoot()->Output_DLR();    tree->Clear();    //test Delete(const Type& )    cout << endl << endl << "Test Delete : " << endl;    tree->Insert(6);    tree->Insert(7);    tree->Insert(9);    tree->Insert(13);    tree->Insert(15);    tree->Insert(4);    tree->Insert(5);    tree->Insert(17);    tree->Insert(19);    tree->Insert(12);    tree->Insert(10);    tree->Insert(14);    tree->Insert(8);    cout << "DLR Output Before Delete: " << endl;    tree->Output_DLR();    tree->Delete(7);    cout << endl << "DLR Output After Delete: " << endl;    tree->Output_DLR();    tree->Clear();return 0;}

测试输出：

对于最后的测试Delete的结果，平衡树的改变可以观察下面这副，数据是相同的