数据结构（C++）-AVL平衡二叉树
2009-04-10 17:54 Iron 阅读(1892) 评论(0) 编辑 收藏 举报类定义文件
//AvlTree.h
#pragma once
//AvlTreeNode平衡二叉树节点
template <class T>
class AvlTreeNode
{
public:
AvlTreeNode<T> *left;
AvlTreeNode<T> *right;
AvlTreeNode<T> *parent;
int balanceFactor;//平衡因子
T data;
AvlTreeNode(const T& item, AvlTreeNode<T>* lptr=NULL, AvlTreeNode<T>* rptr=NULL, AvlTreeNode<T>* par=NULL, int balfac=0);
};
//AvlTree平衡二叉树
template <class T>
class AvlTree
{
public:
AvlTreeNode<T>* root;
AvlTreeNode<T>* current;
int size;
AvlTreeNode<T>* LL(AvlTreeNode<T>* tree);
AvlTreeNode<T>* LR(AvlTreeNode<T>* tree);
AvlTreeNode<T>* RR(AvlTreeNode<T>* tree);
AvlTreeNode<T>* RL(AvlTreeNode<T>* tree);
void buildParent(AvlTreeNode<T>* tree);
bool RotateSubTree(AvlTreeNode<T>* tree);
AvlTree(void);
AvlTreeNode<T>* Search(const T item);
bool Insert(T item);
bool Delete(T item);
void Destory(AvlTreeNode<T>* tree);
~AvlTree(void);
};
类实现文件
//AvlTree.cpp
#include "AvlTree.h"
template <class T>
AvlTreeNode<T>::AvlTreeNode(const T &item, AvlTreeNode<T> *lptr = NULL, AvlTreeNode<T> *rptr = NULL, AvlTreeNode<T>* pr=NULL, int balfac = 0):data(item),left(lptr),right(rptr),parent(pr),balanceFactor(balfac)
{ }
template <class T>
AvlTree<T>::AvlTree():root(NULL),size(0),current(NULL)
{ }
//向平衡二叉树中插入一个元素
template <class T>
bool AvlTree<T>::Insert(const T item)
{
//先找到位置
AvlTreeNode<T>* tempPosition = this->root;
AvlTreeNode<T>* pre = tempPosition;
for(;;)
{
//如果已经到达要插入的地方，则插入新节点
if(tempPosition == NULL)
{
//实例化新节点
AvlTreeNode<T>* newNode = new AvlTreeNode<T>(item);
//如果插入位置为其父节点的左节点，则把新节点挂接到其父节点的左节点上,否则挂接到其父节点的右节点上
if(tempPosition==this->root)
{
root = newNode;
size++;
return true;
}
else if(tempPosition == pre->left && item<pre->data)
{
newNode->parent = pre;
pre->left = tempPosition = newNode;
size++;
break;
}
else
{
newNode->parent = pre;
pre->right = tempPosition = newNode;
size++;
break;
}
}
else if(item < tempPosition->data)
{
//保存tempPosition前一节点
pre = tempPosition;
//在左子数寻找插入点
tempPosition = tempPosition->left;
}
else if(item > tempPosition->data)
{
//保存tempPosition前一节点
pre = tempPosition;
//在右子数寻找插入点
tempPosition = tempPosition->right;
}
else
{
//已存在此元素
return false;
}
}
//调整以使得树平衡
int bf = 0;
while(tempPosition->parent)
{
//bf表示平衡因子的改变量，当新节点插入到左子树，则平衡因子+1
//当新节点插入到右子树，则平衡因子-1
bf = item<tempPosition->parent->data?1:-1;
tempPosition = tempPosition->parent;//将指针指向父节点
tempPosition->balanceFactor += bf;//改变父节点的平衡因子
bf = tempPosition->balanceFactor;//获取当前节点的平衡因子
//判断当前节点平衡因子，如果为表示改子树已平衡，不需要在回溯
//
if(bf==0)
{
return true;
}
else if(bf==2||bf==-2)
{
//调整树以使其平衡
RotateSubTree(tempPosition);
buildParent(root);
root->parent = NULL;
return true;
}
}
return true;
}
//寻找值为item的AvlTree节点
template <class T>
AvlTreeNode<T>* AvlTree<T>::Search(const T item)
{
current = this->root;
for(;;)
{
//current为NULL，没有此节点
if(current==NULL)
return NULL;
//current不为NULL，继续寻找
if(item == current->data)
return current;
else if(item < current->data)
current = current->left;
else if(item > current->data)
current = current->right;
}
}
//删除一个节点
template <class T>
bool AvlTree<T>::Delete(const T item)
{
AvlTreeNode<T>* deleteNode = Search(item);
//如果所删节点不存在，返回
if(deleteNode==NULL)
return false;
//pre在以下的程序中是tempPosition的父节点
AvlTreeNode<T>* tempPosition = NULL;
AvlTreeNode<T>* pre = tempPosition;
//存储真正删除的节点
AvlTreeNode<T>* trueDeleteNode;
//当被删除结点存在左右子树时
if (deleteNode->left != NULL && deleteNode->right != NULL)
{
//获取左子树
tempPosition = deleteNode->left;
//获取deleteNode的中序遍历前驱结点，并存放于tempPosition中
while (tempPosition->right != NULL)
{
//找到左子树中的最右下结点
tempPosition = tempPosition->right;
}
//用中序遍历前驱结点的值代替被删除结点的值
deleteNode->data = tempPosition->data;
if (tempPosition->parent == deleteNode)
{
//如果被删元素的前驱是其左孩子
tempPosition->parent->left = tempPosition->left;
}
else
{
tempPosition->parent->right = tempPosition->left;
}
//得到真正删除的节点
trueDeleteNode = tempPosition;
}
else //当只有左子树或右子树或为叶子结点时
{
//首先找到惟一的孩子结点
pre = deleteNode->parent;
tempPosition = deleteNode->left;
if (tempPosition == NULL) //如果只有右孩子或没孩子
{
tempPosition = deleteNode->right;
}
if (deleteNode!=root)
{
//如果删除节点不是根节点
if (deleteNode->parent->left == deleteNode)
{ //如果被删结点是左孩子
deleteNode->parent->left = tempPosition;
}
else
{ //如果被删结点是右孩子
deleteNode->parent->right = tempPosition;
}
}
else
{
//当删除的是根结点时
root = tempPosition;
}
//得到真正删除的节点
trueDeleteNode = deleteNode;
}
//pre为真正删除节点的父节点
pre = trueDeleteNode==NULL?NULL:trueDeleteNode->parent;
//删除完后进行旋转，现在pre指向实际被删除的结点
while (pre)
{ //bf表示平衡因子的改变量，当删除的是左子树中的结点时，平衡因子-1
//当删除的是右子树的孩子时，平衡因子+1
int bf = (trueDeleteNode->data <= pre->data) ? -1 : 1;
pre->balanceFactor += bf; //改变当父结点的平衡因子
tempPosition = pre;
pre = pre->parent;
bf = tempPosition->balanceFactor; //获取当前结点的平衡因子
if (bf != 0) //如果bf==0，表明高度降低，继续后上回溯
{
//如果bf为或-1则说明高度未变，停止回溯，如果为或-2，则进行旋转
//当旋转后高度不变，则停止回溯
if (bf == 1 || bf == -1 || !RotateSubTree(tempPosition))
{
break;
}
}
}
buildParent(root);
if(root!=NULL)
root->parent = NULL;
delete trueDeleteNode;//析构真正删除的节点
size--;
return true;
}
//调整函数
template <class T>
bool AvlTree<T>::RotateSubTree(AvlTreeNode<T>* tree)
{
this->current = tree;
bool tallChange = true;
int bf = tree->balanceFactor;
AvlTreeNode<T>* newRoot = NULL;
if (bf == 2) //当平衡因子为时需要进行旋转操作
{
int leftBF = current->left->balanceFactor;//得到左子树的平衡因子
if (leftBF == -1)
{
newRoot = LR(tree);//LR型旋转(左子树中插入右孩子，先左后右双向旋转)
}
else if (leftBF == 1)
{
newRoot = LL(tree); //LL型旋转(左子树中插入左孩子，右单转)
}
else //当旋转根左孩子的bf为时，只有删除时才会出现
{
newRoot = LL(tree);
tallChange = false;
}
}
if (bf == -2) //当平衡因子为-2时需要进行旋转操作
{
int rightBF = current->right->balanceFactor; //获取旋转根右孩子的平衡因子
if (rightBF == 1)
{
newRoot = RL(tree); //RL型旋转(右子树中插入左孩子，先右后左双向旋转)
}
else if (rightBF == -1)
{
newRoot = RR(tree); //RR型旋转(右子树中插入右孩子，左单转)
}
else //当旋转根左孩子的bf为时，只有删除时才会出现
{
newRoot = RR(tree);
tallChange = false;
}
}
//更改新的子树根
if (current->parent!=NULL)
{
//如果旋转根为不是AVL树的根
//newRoot为新旋转后得到的当前子树的根，一下判断的作用是
//如果原来的根是其父节点的左子树，则新根也挂接到其父节点的左子树，
//否则挂接到右子树
if (tree->data < (tree->parent)->data)
{
current->parent->left = newRoot;
}
else
{
current->parent->right = newRoot;
}
}
else
{
//如果旋转根为AVL树的根，则指定新AVL树根结点
this->root = newRoot;
}
return tallChange;
}
//LL型旋转，返回旋转后的新子树根
template <class T>
AvlTreeNode<T>* AvlTree<T>::LL(AvlTreeNode<T>* tree)
{
AvlTreeNode<T>* treeNext = tree->left;
tree->left = treeNext->right;
treeNext->right = tree;
if (treeNext->balanceFactor == 1)
{
tree->balanceFactor = 0;
treeNext->balanceFactor = 0;
}
else //treeNext->balanceFactor==0的情况，删除时用
{
tree->balanceFactor = 1;
treeNext->balanceFactor = -1;
}
return treeNext; //treeNext为新子树的根
}
//LR型旋转，返回旋转后的新子树根
template <class T>
AvlTreeNode<T>* AvlTree<T>::LR(AvlTreeNode<T>* tree)
{
AvlTreeNode<T>* treeNext = tree->left;
AvlTreeNode<T>* newRoot = treeNext->right;
tree->left = newRoot->right;
treeNext->right = newRoot->left;
newRoot->left = treeNext;
newRoot->right = tree;
switch (newRoot->balanceFactor) //改变平衡因子
{
case 0:
tree->balanceFactor = 0;
treeNext->balanceFactor = 0;
break;
case 1:
tree->balanceFactor = -1;
treeNext->balanceFactor = 0;
break;
case -1:
tree->balanceFactor = 0;
treeNext->balanceFactor = 1;
break;
}
newRoot->balanceFactor = 0;
return newRoot; //newRoot为新子树的根
}
//RR型旋转，返回旋转后的新子树根
template <class T>
AvlTreeNode<T>* AvlTree<T>::RR(AvlTreeNode<T>* tree)
{
AvlTreeNode<T>* treeNext = tree->right;
tree->right = treeNext->left;
treeNext->left = tree;
if (treeNext->balanceFactor == -1)
{
tree->balanceFactor = 0;
treeNext->balanceFactor = 0;
}
else //treeNext->balanceFactor==0的情况，删除时用
{
tree->balanceFactor = -1;
treeNext->balanceFactor = 1;
}
return treeNext; //treeNext为新子树的根
}
//RL型旋转，返回旋转后的新子树根
template <class T>
AvlTreeNode<T>* AvlTree<T>::RL(AvlTreeNode<T>* tree)
{
AvlTreeNode<T>* treeNext = tree->right;
AvlTreeNode<T>* newRoot = treeNext->left;
tree->right = newRoot->left;
treeNext->left = newRoot->right;
newRoot->right = treeNext;
newRoot->left = tree;
switch (newRoot->balanceFactor) //改变平衡因子
{
case 0:
tree->balanceFactor = 0;
treeNext->balanceFactor = 0;
break;
case 1:
tree->balanceFactor = 0;
treeNext->balanceFactor = -1;
break;
case -1:
tree->balanceFactor = 1;
treeNext->balanceFactor = 0;
break;
}
newRoot->balanceFactor = 0;
return newRoot; //newRoot为新子树的根
}
//重新建立所有节点的父节点连接
template <class T>
void AvlTree<T>::buildParent(AvlTreeNode<T>* tree)
{
//如果树为空则直接返回，如果树有节点的parent置为当前tree节点
if(tree==NULL)
{
return;
}
if(tree->left!=NULL)
{
tree->left->parent = tree;
}
if(tree->right!=NULL)
{
tree->right->parent = tree;
}
//递归遍历子节点建立父节点连接
buildParent(tree->left);
buildParent(tree->right);
}
//销毁平衡树
template <class T>
void AvlTree<T>::Destory(AvlTreeNode<T>* tree)
{
//如果树为空
if(tree==NULL)
return;
//如果树不为空
if(tree->left!=NULL || tree->right!=NULL)
{
//析构左子树
if(tree->left!=NULL)
{
Destory(tree->left);
tree->left = NULL;
}
//析构右子树
if(tree->right!=NULL)
{
Destory(tree->right);
tree->right = NULL;
}
}
//析构树
delete tree;
}
template <class T>
AvlTree<T>::~AvlTree(void)
{
Destory(root);
}
测试文件//avlmain.cpp
//#include <vld.h>//测试内存泄漏工具
#include <iostream>
#include "AvlTree.cpp"
using namespace std;
//主函数测试
int main()
{
AvlTree<int>* intAvlTree = new AvlTree<int>();
intAvlTree->Insert(10);
intAvlTree->Insert(14);
intAvlTree->Insert(9);
intAvlTree->Insert(13);
intAvlTree->Insert(15);
intAvlTree->Insert(12);
intAvlTree->Insert(11);
intAvlTree->Delete(13);
intAvlTree->Delete(11);
intAvlTree->Delete(14);
intAvlTree->Delete(10);
intAvlTree->Delete(9);
intAvlTree->Delete(15);
intAvlTree->Delete(12);
//cout<<intAvlTree->Search(11)->data;
delete intAvlTree;
return 1;
}
最近在重新看数据结构，偶然看到AVL树，以前没有实现过，现在有空，就写了一个，希望大家批评指正。
参考资料：
1.http://www.cnblogs.com/abatei/archive/2008/11/17/1335031.html
这个资料里面的旋转讲的非常好，我的avl实现基本在此文章提供的代码上修改
2.《数据结构 基于c++模板类的实现》作者：余腊生
看懂avl的原理,此书的讲解功不可没
