平衡二叉树( AVL 树)
在上面一章中我们介绍了二叉排序树，它是一种既方便查找也有利于插入和删除的方法。但是当输入的排好序数据组时，二叉排序树为斜树，不利于查找效率的稳定性差。现在我们介绍一种高度平衡的二叉排序树——平衡二叉树( AVL 树)。
平衡二叉树，是一种二叉排序树，其中每一个节点的左子树和右子树的高度差至多等于1。
将二叉树上结点的左子树深度减去右子树深度的值称为平衡因子BF (Balance Factor) 。我们来根据上一章的代码进行改进：
#include<vector>
#define LH +1//左高
#define EH 0//等高
#define RH -1//右高
using namespace std;
class BinSortT
{
public:
BinSortT();
~BinSortT();
friend void CreateBSTree(vector<int>t, int k, BinSortT* T, BinSortT *P);//根据一组数据创建二叉排序树
friend bool SearchBST(BinSortT* T, int key);//查找
friend bool InsertBST(BinSortT* T, int key);//查找不成功插入
friend bool DeleteBST(BinSortT* T, int key);//删除数据
friend bool Delete(BinSortT* T);//删除操作
friend bool InsertAVL(BinSortT* T, int e, bool *taller, int &f);//插入元素
friend void R_Rotate(BinSortT* T);//右旋处理
friend void L_Rotate(BinSortT* T);//左旋处理
friend void LeftBalance(BinSortT* T);//左平衡旋转处理
friend void RightBalance(BinSortT* T);//右平衡旋转处理
int data;//结点数据
int bf; //结点的平衡因子
private:
BinSortT*lchild, *rchild;//左右孩子指针
};
#include "BinSortT.h"
BinSortT::BinSortT()
{
}
BinSortT::~BinSortT()
{
}
void R_Rotate(BinSortT* T)//右旋处理
{
BinSortT *L = new BinSortT;
*L = *T->lchild;
T->lchild = L->rchild;
BinSortT *p = new BinSortT;
*p = *T;
L->rchild = p;
*T = *L;
delete L;
}
void L_Rotate(BinSortT* T)//左旋处理
{
BinSortT *R = new BinSortT;
*R = *T->rchild;
T->rchild =R->lchild;
BinSortT *p = new BinSortT;
*p = *T;
R->lchild = p;
*T= *R;
delete R;
}
void LeftBalance(BinSortT* T)//左平衡旋转处理
{
BinSortT *L, *Lr;
L = T->lchild;
switch(T->bf)
{
case LH:
T->bf = EH;
L->bf = EH;
R_Rotate(T);
break;
case EH:
T->bf = LH;
L->bf = RH;
R_Rotate(T);
break;
case RH:
Lr = T->rchild;
switch (Lr->bf)
{
case LH:
T->bf = RH;
L->bf = EH;
break;
case EH:
T->bf = EH;
L->bf = EH;
break;
case RH:
T->bf = EH;
L->bf = LH;
break;
}
Lr->bf = EH;
L_Rotate(T);
R_Rotate(T);
}
}
void RightBalance(BinSortT* T)//右平衡旋转处理
{
BinSortT *R, *Rl;
R = T->rchild;
switch (T->bf)
{
case RH:
T->bf = EH;
R->bf = EH;
L_Rotate(T);
break;
case EH:
T->bf = RH;
R->bf = LH;
L_Rotate(T);
break;
case LH:
Rl = T->lchild;
switch (Rl->bf)
{
case RH:
T->bf = LH;
R->bf = EH;
break;
case EH:
T->bf = EH;
R->bf = EH;
break;
case LH:
T->bf = EH;
R->bf = RH;
break;
}
Rl->bf = EH;
R_Rotate(T);
L_Rotate(T);
}
}
#include <iostream>
#include "BinSortT.h"
using namespace std;
int main()
{
int e,n,f=0;
bool *taller=new bool;
*taller = true;
vector<int> vec;
cout << "输入数据元素个数：";
cin >> n;
for (int i = 0; i < n; i++)
{
cin >> e;
vec.push_back(e);
}
BinSortT *T = new BinSortT;
for (int i = 0; i < n; i++)
{
InsertAVL(T, vec[i], taller,f);
}
system("pause");
return 0;
}
void CreateBSTree(vector<int>t, int k, BinSortT *T, BinSortT *P)//根据一组数据创建二叉排序树
{
T->data = t[k++];
while (k < t.size())
{
if (t[k] < P->data)
{
if (!P->lchild)
{
P->lchild = new BinSortT;
P->lchild->data = t[k++];
P = T;
}
else
{
P = P->lchild;
}
}
else if (t[k] > P->data)
{
if (!P->rchild)
{
P->rchild = new BinSortT;
P->rchild->data = t[k++];
P = T;
}
else
{
P = P->rchild;
}
}
}
}
bool SearchBST(BinSortT* T, int key)//查找
{
if (!T)
return false;
if (key == T->data)
return true;
if (key < T->data)
SearchBST(T->lchild, key);
else
SearchBST(T->rchild, key);
}
bool InsertBST(BinSortT* T, int key)//查找不成功插入
{
if (!T)
{
T = new BinSortT;
T->data = key;
return false;
}
while (T)
{
if (key == T->data)return true;
if (key < T->data)
{
if (!T->lchild)
{
T->lchild = new BinSortT;
T->lchild->data = key;
return false;
}
else
{
T = T->lchild;
}
}
else if (key > T->data)
{
if (!T->rchild)
{
T->rchild = new BinSortT;
T->rchild->data = key;
return false;
}
else
{
T = T->rchild;
}
}
}
}
bool DeleteBST(BinSortT* T, int key)//删除数据
{
if (!T)return false;
if (key == T->data)
return Delete(T);
else if (key < T->data)
return DeleteBST(T->lchild, key);
else if (key >T->data)
return DeleteBST(T->rchild, key);
}
bool Delete(BinSortT* T)//删除操作
{
BinSortT *q, *s;
if (T->rchild == NULL)
{
q = T->lchild;
*T = *(T->lchild);
delete q;
}
else if (T->lchild == NULL)
{
q = T->rchild;
*T = *(T->rchild);
delete q;
}
else
{
s = T->lchild;
q = T;
while (s->rchild)
{
q = s;
s = s->rchild;
}
T->data = s->data;
if (q != T)
q->rchild = s->lchild;
else
q->lchild = s->lchild;
delete s;
}
return true;
}
bool InsertAVL(BinSortT* T, int e, bool *taller,int &f)//插入元素
{
if (!f)
{
T->data = e;
T->bf = EH;
T->lchild = T->rchild = NULL;
*taller = true;
f = 1;
}
else
{
if (e == T->data)
{
*taller = false;
return false;
}
if (e < T->data)
{
if (!T->lchild)
{
T->lchild = new BinSortT;
f = 0;
}
if (!InsertAVL(T->lchild, e, taller,f))
return false;
if (*taller)
{
switch (T->bf)
{
case LH:
LeftBalance(T);
*taller = false;
break;
case EH:
T->bf = LH;
*taller = true;
break;
case RH:
T->bf = EH;
*taller = false;
break;
}
}
}
else
{
if (!T->rchild)
{
T->rchild = new BinSortT;
f = 0;
}
if (!InsertAVL(T->rchild, e, taller,f))
return false;
if (*taller)
{
switch (T->bf)
{
case RH:
RightBalance(T);
*taller = false;
break;
case EH:
T->bf = RH;
*taller = true;
break;
case LH:
T->bf = EH;
*taller = false;
break;
}
}
}
}
return true;
}
在VS上运行结果如下：
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