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
BSTNode* AVLTree::LL(BSTNode* &topNode){
BSTNode * 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
BSTNode* AVLTree::RR(BSTNode* &topNode){
BSTNode *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
BSTNode* AVLTree::LR(BSTNode* &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
BSTNode* AVLTree::RL(BSTNode* &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
BSTNode* AVLTree::Delete(const Type& key){
return root = Delete(root, key);
}
template
BSTNode* AVLTree::Delete(BSTNode* &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 *newNode = node->left;
delete node;
return newNode;
}else{
BSTNode *secondMostLeftNode = node->right;
if(secondMostLeftNode->left == NULL){
return secondMostLeftNode;
}
while(secondMostLeftNode->left->left)
secondMostLeftNode = secondMostLeftNode->left;
BSTNode *mostLeftNode = secondMostLeftNode->left;
secondMostLeftNode->left->left = node->left;
secondMostLeftNode->left->right = node->right;
secondMostLeftNode->left = NULL;
return mostLeftNode;
}
}
//from bottom to the top
else 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
BSTNode* AVLTree::Rotate(BSTNode* node){
if(node->left->getHeight() - node->right->getHeight() == 2){
if(node->left->left->getHeight() >= node->left->right->getHeight())
node = LL(node);
else
node = LR(node);
}
if(node->right->getHeight() - node->left->getHeight() == 2){
if(node->right->right->getHeight() >= node->right->left->getHeight())
node = RR(node);
else
node = 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
#include
#include
using namespace std;
template
class AVLTree;
/**
* Binary Search Tree Node: BSTNode class
*/
template
class BSTNode{
friend class AVLTree;
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 order
if(this != NULL){
cout << key << ", ";
left->Output_DLR();
right->Output_DLR();
}
}
};
/**
* AVLTree class
*/
template
class AVLTree{
private:
BSTNode *root;
public:
AVLTree(): root(NULL){}
BSTNode* Insert(BSTNode* &, const Type&);
BSTNode* Insert(const Type& );
BSTNode* Delete(BSTNode* &, const Type&);
BSTNode* Delete(const Type& );
BSTNode* Rotate(BSTNode* );
BSTNode* GetRoot();
BSTNode* LL(BSTNode* &);
BSTNode* LR(BSTNode* &);
BSTNode* RL(BSTNode* &);
BSTNode* RR(BSTNode* &);
void Clear();
void Output_DLR();
void Output_LRN();
};
template
BSTNode* AVLTree::LL(BSTNode* &topNode){
BSTNode * leftSonNode = topNode->left;
topNode->left = leftSonNode->right;
leftSonNode->right = topNode;
return leftSonNode;
}
template
BSTNode* AVLTree::RR(BSTNode* &topNode){
BSTNode *rightSonNode = topNode->right;
topNode->right = rightSonNode->left;
rightSonNode->left = topNode;
return rightSonNode;
}
template
BSTNode* AVLTree::LR(BSTNode* &topNode){
topNode->left = RR(topNode->left);
return LL(topNode);
}
template
BSTNode* AVLTree::RL(BSTNode* &topNode){
topNode->right = LL(topNode->right);
return RR(topNode);
}
template
BSTNode* AVLTree::GetRoot(){
return root;
}
template
BSTNode* AVLTree::Insert(const Type& key){
return Insert(root, key);
}
template
BSTNode* AVLTree::Insert(BSTNode* &node, const Type &key){
if(node == NULL){
return node = new BSTNode(key);
}
//from bottom to the top
else if(key < node->key){
Insert(node->left, key);
if(node->left->getHeight() - node->right->getHeight() == 2){
if(key < node->left->key)
node = LL(node);
else
node = LR(node);
}
}
else{
Insert(node->right, key);
if(node->right->getHeight() - node->left->getHeight() == 2){
if(key > node->right->key)
node = RR(node);
else
node = RL(node);
}
}
return node;
}
template
BSTNode* AVLTree::Delete(const Type& key){
return root = Delete(root, key);
}
template
BSTNode* AVLTree::Delete(BSTNode* &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 *newNode = node->left;
delete node;
return newNode;
}else{
BSTNode *secondMostLeftNode = node->right;
if(secondMostLeftNode->left == NULL){
return secondMostLeftNode;
}
while(secondMostLeftNode->left->left)
secondMostLeftNode = secondMostLeftNode->left;
BSTNode *mostLeftNode = secondMostLeftNode->left;
secondMostLeftNode->left->left = node->left;
secondMostLeftNode->left->right = node->right;
secondMostLeftNode->left = NULL;
return mostLeftNode;
}
}
//from bottom to the top
else 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
BSTNode* AVLTree::Rotate(BSTNode* node){
if(node->left->getHeight() - node->right->getHeight() == 2){
if(node->left->left->getHeight() >= node->left->right->getHeight())
node = LL(node);
else
node = LR(node);
}
if(node->right->getHeight() - node->left->getHeight() == 2){
if(node->right->right->getHeight() >= node->right->left->getHeight())
node = RR(node);
else
node = RL(node);
}
return node;
}
template
void AVLTree::Clear(){
root->clear();
root = NULL;
}
template
void AVLTree::Output_DLR(){
if(!root)
cout << "EMPTY TREE! " << endl;
else
root->Output_DLR();
}
template
void AVLTree::Output_LRN(){
if(!root)
cout << "EMPTY TREE! " << endl;
else
root->Output_LRN();
}
//Test Main
int main() {
AVLTree *tree = new AVLTree();
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