一棵树是否是AVL,则只要这棵二叉树满足是BST并且每个结点的平衡因子都满足在1、0、-1的范围。则符合条件。
核心代码如下:
template <typename keyType>
bool is_AVL(BT<keyType> &bt){
return is_AVL_Core(bt.root);
}
template <typename keyType>
bool is_AVL_Core(BinaryTreeNode<keyType> *root){
return is_BST(root,INT_MIN,INT_MAX) && is_balance(root);
}
template <typename keyType>
bool is_BST(BinaryTreeNode<keyType> *root,keyType min,keyType max){//判断是否是BST
if(root == nullptr)
return true;
if(root->value < min || root->value > max)
return false;
return is_BST(root->m_pLeft,min,root->value - 1) && is_BST(root->m_pRight,root->value + 1,max);
}
template <typename keyType>
bool is_balance(BinaryTreeNode<keyType> *root){//判断二叉树平衡因子是否满足
if(root == nullptr)
return true;
if(BT<keyType>::diff(root) > 1 || BT<keyType>::diff(root) < -1)
return false;
return is_balance(root->m_pLeft) && is_balance(root->m_pRight);
}template <typename keyType>
int BT<keyType>::getHeight(){
_getHeight(root);
}
template <typename keyType>
int BT<keyType>::_getHeight(BTNode *root){//获取结点高度
if(root == nullptr)
return 0;
return max(_getHeight(root->m_pLeft),_getHeight(root->m_pRight))+1;
}
template <typename keyType>
int BT<keyType>::diff(BTNode *root){//计算结点平衡因子
if(root == nullptr)
return 0;
return _getHeight(root->m_pLeft) - _getHeight(root->m_pRight);
}程序完整代码如下:
#include<iostream>
#include<algorithm>
#include<queue>
using namespace std;
template <typename keyType>//之后的类名一定要记得加上<typename>
class BinaryTreeNode{
public:
keyType value;
BinaryTreeNode *m_pLeft;
BinaryTreeNode *m_pRight;
BinaryTreeNode(keyType v):value(v),m_pLeft(nullptr),m_pRight(nullptr){}
};
//AVL树
template <typename keyType>
class BT{
typedef BinaryTreeNode<keyType> BTNode;//给结点定义别名
public:
BT(){ root = nullptr; }//默认构造函数
BTNode * insertBTNode(BTNode *&root,keyType val);
BTNode *createBT(keyType arr[],int n);//通过数组创建二叉排序树,通过*&来改变指针的值
BTNode *createBT_ByQue(queue<keyType> que);//通过队列创建二叉排序树
BTNode *search_AVL(keyType v);
BTNode *search_AVL_Core(BTNode *root,keyType v);
void midorder_showBT();//中序遍历
void midorder_showBT_core(const BTNode *root);
void level_showBT();//中序遍历
void level_showBT_core(BTNode *root);
~BT();//析构函数
void release_BT_core(BTNode *root);
static int _getHeight(BTNode *root);
int getHeight();//获取高度
static int diff(BTNode *root);//计算平衡因子
static BTNode *balance(BTNode *root);//平衡操作
static BTNode *LL_Rotation(BTNode *root);//LL旋转
static BTNode *LR_Rotation(BTNode *root);
static BTNode *RL_Rotation(BTNode *root);
static BTNode *RR_Rotation(BTNode *root);
queue<keyType> get_AVL_Node();
void get_AVL_Node_Core(queue<keyType> &que,BTNode *root);
BTNode *root;
};
int INT_MIN = -99999,INT_MAX = 99999;
template <typename keyType>
//注意这里要是引用，不然析构时会出错
bool is_AVL(BT<keyType> &bt);//判断是否是平衡二叉树
template <typename keyType>
bool is_AVL_Core(BinaryTreeNode<keyType> *root);
template <typename keyType>
bool is_BST(BinaryTreeNode<keyType> *root,keyType min,keyType max);
template <typename keyType>
bool is_balance(BinaryTreeNode<keyType> *root);
int main(){
BT<int> bt;
int arr[] = {16,3,7,11,9,26,18,14,15};
bt.createBT(arr,sizeof(arr)/sizeof(arr[0]));//构建二叉排序树
cout<<"中序遍历:"<<endl;
bt.midorder_showBT();//中序遍历打印
cout<<endl<<endl;
cout<<"层次遍历:"<<endl;
bt.level_showBT();//前序遍历打印
cout<<endl<<endl;
if(is_AVL(bt))//可通过把构建AVL树过程中的balance函数注释掉，则看到不是AVL树
cout<<"是平衡二叉树"<<endl;
else
cout<<"不是平衡二叉树"<<endl;
return 0;
}
template <typename keyType>
bool is_AVL(BT<keyType> &bt){
return is_AVL_Core(bt.root);
}
template <typename keyType>
bool is_AVL_Core(BinaryTreeNode<keyType> *root){
return is_BST(root,INT_MIN,INT_MAX) && is_balance(root);
}
template <typename keyType>
bool is_BST(BinaryTreeNode<keyType> *root,keyType min,keyType max){//判断是否是BST
if(root == nullptr)
return true;
if(root->value < min || root->value > max)
return false;
return is_BST(root->m_pLeft,min,root->value - 1) && is_BST(root->m_pRight,root->value + 1,max);
}
template <typename keyType>
bool is_balance(BinaryTreeNode<keyType> *root){//判断二叉树平衡因子是否满足
if(root == nullptr)
return true;
if(BT<keyType>::diff(root) > 1 || BT<keyType>::diff(root) < -1)
return false;
return is_balance(root->m_pLeft) && is_balance(root->m_pRight);
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::createBT(keyType arr[],int n){
root = nullptr;
for(int i=0;i<n;i++)
insertBTNode(root,arr[i]);
return root;
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::createBT_ByQue(queue<keyType> que){
root = nullptr;
while(!que.empty()){
insertBTNode(root,que.front());
que.pop();
}
return root;
}
template <typename keyType>
BinaryTreeNode<keyType> * BT<keyType>::insertBTNode(BTNode *&root,keyType val){
if(root == nullptr){
root = new BTNode(val);
return root;
}
if(val == root->value){
return root;
}
else if(val < root->value){
root->m_pLeft = insertBTNode(root->m_pLeft,val);
root = balance(root); //注意这里是把平衡后的返回置temp赋值给root
return root;
}
else{
root->m_pRight = insertBTNode(root->m_pRight,val);
root = balance(root); //注意这里是把平衡后的返回置temp赋值给root
return root;
}
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::balance(BTNode *root){
int dis = diff(root);//计算结点的平衡因子
if(dis > 1){//左
if( diff(root->m_pLeft) > 0)
return LL_Rotation(root);
else
return LR_Rotation(root);
}
else if(dis < -1){//右
if( diff(root->m_pRight) < 0)
return RR_Rotation(root);
else
return RL_Rotation(root);
}
return root;//无需转换时记得返回root
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::LL_Rotation(BTNode *root){
BTNode *temp = root->m_pLeft;
root->m_pLeft = temp->m_pRight;
temp->m_pRight = root;
return temp;//返回要旋转子树的主结点
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::RR_Rotation(BTNode *root){
BTNode *temp = root->m_pRight;
root->m_pRight = temp->m_pLeft;
temp->m_pLeft = root;
return temp;
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::LR_Rotation(BTNode *root){//先进行RR操作，再进行LL操作
//注意这里一定要对root->m_Pleft重新赋值
root->m_pLeft = RR_Rotation(root->m_pLeft);//先对root后的左结点进行RR操作
return LL_Rotation(root);//再对root进行LL操作
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::RL_Rotation(BTNode *root){//先进行LL操作，再进行RR操作
//注意这里一定要对root->m_pRight重新赋值
root->m_pRight = LL_Rotation(root->m_pRight);//先对root后的右结点进行LL操作
return RR_Rotation(root);//再对root进行RR操作
}
template <typename keyType>
void BT<keyType>::midorder_showBT(){
midorder_showBT_core(root);
}
template <typename keyType>
void BT<keyType>::midorder_showBT_core(const BTNode *root){
if(root == nullptr){
return;
}
midorder_showBT_core(root->m_pLeft);
cout<<root->value<<" ";
midorder_showBT_core(root->m_pRight);
}
template <typename keyType>
void BT<keyType>::level_showBT()//中序遍历
{
level_showBT_core(root);
}
template <typename keyType>
void BT<keyType>::level_showBT_core(BTNode *root){
if(root == nullptr)
return;
queue<BinaryTreeNode<keyType> *> que;
que.push(root);
while(!que.empty()){
if( que.front()->m_pLeft != nullptr)
que.push(que.front()->m_pLeft);
if(que.front()->m_pRight != nullptr)
que.push(que.front()->m_pRight);
cout<<que.front()->value<<" ";
que.pop();
}
}
template <typename keyType>
BT<keyType>::~BT()//释放二叉树
{
release_BT_core(root);
}
template <typename keyType>
void BT<keyType>::release_BT_core(BTNode *root)//释放二叉树
{
if(root == nullptr)
return;
release_BT_core(root->m_pLeft);
release_BT_core(root->m_pRight);
delete root;
root = nullptr;
return ;
}
template <typename keyType>
int BT<keyType>::getHeight(){
_getHeight(root);
}
template <typename keyType>
int BT<keyType>::_getHeight(BTNode *root){//获取结点高度
if(root == nullptr)
return 0;
return max(_getHeight(root->m_pLeft),_getHeight(root->m_pRight))+1;
}
template <typename keyType>
int BT<keyType>::diff(BTNode *root){//计算结点平衡因子
if(root == nullptr)
return 0;
return _getHeight(root->m_pLeft) - _getHeight(root->m_pRight);
}
template <typename keyType>
queue<keyType> BT<keyType>::get_AVL_Node()//获取AVL树的所有结点值并返回
{
queue<keyType> que;
get_AVL_Node_Core(que,root);
return que;
}
template <typename keyType>
void BT<keyType>::get_AVL_Node_Core(queue<keyType> &que,BTNode *root)//获取AVL树的所有结点值并返回
{
if(root == nullptr)
return ;
get_AVL_Node_Core(que,root->m_pLeft);
que.push(root->value);
get_AVL_Node_Core(que,root->m_pRight);
}
//查找结点，AVLTree查找的复杂度能控制在对数范围O(log n)
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::search_AVL(keyType v){
return search_AVL_Core(root,v);
}
template <typename keyType>
BinaryTreeNode<keyType> *BT<keyType>::search_AVL_Core(BTNode *root,keyType v){
if(root == nullptr)
return nullptr;
if(v == root->value)
return root;
else if(v < root->value)
search_AVL_Core(root->m_pLeft,v);
else
search_AVL_Core(root->m_pRight,v);
}运行结果如下:
