1、AVL树:

    1)其左子树(TL)与右子树(TR)是AVL树;

    2)|HL-HR|<=1,其中HL和HR是TL和TR的高度;

    3)高度为h的AVL树,结点数2*h-1。

    AVL树查找,插入,删除在平均和最坏情况下都是O(logn),插入和删除可能需要一次或多次旋转重新达到平衡。AVL树的旋转平衡思路:以不平衡点为根的子树高度应保持不变,新结点插入后,向根回溯到第一个原平衡因  子不为0的结点。旋转方法如下:

  1)LL型:左旋转

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" alt="" />

  2)RR型:右旋转

    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" 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  3)LR型:在不平衡的儿结点先进行右旋转,然后进行左旋转。

    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" alt="" />

  4)RL型:在不平衡的儿结点先进行左旋转,然后里德右旋转

    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" alt="" />

  注意在旋转过程中涉及的最小树应该保持搜索二叉树的性质。

  AVL树的C++实现核心部分是平衡旋转:

 #include "stdafx.h"
#include<time.h>
#include<stdlib.h>
#include<iostream>
using namespace std; typedef struct AVLTree{
int ndata;
AVLTree* pLchild;
AVLTree* pRchild;
int nheight;
}AVLTree; AVLTree* LLRotate(AVLTree* pRoot);
AVLTree* RRRotate(AVLTree* pRoot);
AVLTree* LRRotate(AVLTree* pRoot);
AVLTree* RLRotate(AVLTree* pRoot); int Compare(int a,int b){
return(a > b ? a : b);
}
int Height(AVLTree* pRoot){
if (NULL==pRoot)
{
return -;
}
else
{
return(pRoot->nheight);
}
} AVLTree* Insert(AVLTree* pRoot, int nData){
if (NULL==pRoot)
{
pRoot = new AVLTree;
pRoot->ndata = nData;
pRoot->nheight = ;
pRoot->pLchild = NULL;
pRoot->pRchild = NULL;
}
else if (pRoot->ndata>nData)
{
pRoot->pLchild = Insert(pRoot->pLchild, nData);
if (Height(pRoot->pLchild) - Height(pRoot->pRchild)==)
{
if (pRoot->pLchild->ndata>nData)
{
pRoot = LLRotate(pRoot);
}
else
{
pRoot = LRRotate(pRoot);
}
}
}
else if (pRoot->ndata<nData)
{
pRoot->pRchild = Insert(pRoot->pRchild, nData);
if (Height(pRoot->pRchild) - Height(pRoot->pLchild) == )
{
if (pRoot->pRchild->ndata<nData)
{
pRoot = RRRotate(pRoot);
}
else
{
pRoot = RLRotate(pRoot);
}
}
}
pRoot->nheight = Compare(Height(pRoot->pLchild), Height(pRoot->pRchild)) + ;
return pRoot;
} //LL旋转
AVLTree* LLRotate(AVLTree* pRoot){
AVLTree* pTemp; pTemp = pRoot->pLchild;
pRoot->pLchild = pTemp->pRchild;
pTemp->pRchild = pRoot; pRoot->nheight = Compare(Height(pRoot->pLchild), Height(pRoot->pRchild)) + ;
pTemp->nheight = Compare(Height(pTemp->pLchild), pRoot->nheight) + ;
return pTemp;
} //RR旋转
AVLTree* RRRotate(AVLTree* pRoot){
AVLTree* pTemp; pTemp = pRoot->pRchild;
pRoot->pRchild = pTemp->pLchild;
pTemp->pLchild = pRoot; pRoot->nheight = Compare(Height(pRoot->pLchild), Height(pRoot->pRchild)) + ;
pTemp->nheight = Compare(Height(pTemp->pRchild), pRoot->nheight) + ;
return pTemp;
} //LR旋转
AVLTree* LRRotate(AVLTree* pRoot){
pRoot->pLchild = RRRotate(pRoot->pLchild);
return LLRotate(pRoot);
} //RL旋转
AVLTree* RLRotate(AVLTree* pRoot){
pRoot->pRchild = LLRotate(pRoot->pRchild);
return RRRotate(pRoot);
} //输出树
void PrintTree(AVLTree* pRoot){
if (NULL==pRoot)
{
return;
}
static int n = ;
PrintTree(pRoot->pLchild);
cout << ++n << "\t" << pRoot->ndata << "\t" << pRoot->nheight << "\n";
PrintTree(pRoot->pRchild);
} int main()
{
AVLTree* pRoot = NULL;
srand((unsigned int)time(NULL));
for (int i = ; i < ; i++)
{
pRoot = Insert(pRoot, rand() % );
}
PrintTree(pRoot);
return ;
}

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