平衡二叉树
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#include <iostream>
#include <algorithm>
using namespace std;
//平衡二叉树结点
template <typename T>
struct AvlNode
{
T data;
int height; //结点所在高度
AvlNode<T> *left;
AvlNode<T> *right;
AvlNode<T>(const T theData) : data(theData), left(NULL), right(NULL), height(0){}
};
//AvlTree
template <typename T>
class AvlTree
{
public:
AvlTree<T>(){}
~AvlTree<T>(){}
AvlNode<T> *root;
//插入结点
void Insert(AvlNode<T> *&t, T x);
//删除结点
bool Delete(AvlNode<T> *&t, T x);
//查找是否存在给定值的结点
bool Contains(AvlNode<T> *t, const T x) const;
//中序遍历
void InorderTraversal(AvlNode<T> *t);
//前序遍历
void PreorderTraversal(AvlNode<T> *t);
//最小值结点
AvlNode<T> *FindMin(AvlNode<T> *t) const;
//最大值结点
AvlNode<T> *FindMax(AvlNode<T> *t) const;
private:
//求树的高度
int GetHeight(AvlNode<T> *t);
//单旋转 左
AvlNode<T> *LL(AvlNode<T> *t);
//单旋转 右
AvlNode<T> *RR(AvlNode<T> *t);
//双旋转 右左
AvlNode<T> *LR(AvlNode<T> *t);
//双旋转 左右
AvlNode<T> *RL(AvlNode<T> *t);
};
template <typename T>
AvlNode<T> * AvlTree<T>::FindMax(AvlNode<T> *t) const
{
if (t == NULL)
return NULL;
if (t->right == NULL)
return t;
return FindMax(t->right);
}
template <typename T>
AvlNode<T> * AvlTree<T>::FindMin(AvlNode<T> *t) const
{
if (t == NULL)
return NULL;
if (t->left == NULL)
return t;
return FindMin(t->left);
}
template <typename T>
int AvlTree<T>::GetHeight(AvlNode<T> *t)
{
if (t == NULL)
return -1;
else
return t->height;
}
//单旋转
//左左插入导致的不平衡
template <typename T>
AvlNode<T> * AvlTree<T>::LL(AvlNode<T> *t)
{
AvlNode<T> *q = t->left;
t->left = q->right;
q->right = t;
t = q;
t->height = max(GetHeight(t->left), GetHeight(t->right)) + 1;
q->height = max(GetHeight(q->left), GetHeight(q->right)) + 1;
return q;
}
//单旋转
//右右插入导致的不平衡
template <typename T>
AvlNode<T> * AvlTree<T>::RR(AvlNode<T> *t)
{
AvlNode<T> *q = t->right;
t->right = q->left;
q->left = t;
t = q;
t->height = max(GetHeight(t->left), GetHeight(t->right)) + 1;
q->height = max(GetHeight(q->left), GetHeight(q->right)) + 1;
return q;
}
//双旋转
//插入点位于t的左儿子的右子树
template <typename T>
AvlNode<T>* AvlTree<T>::LR(AvlNode<T> *t)
{
//双旋转可以通过两次单旋转实现
//对t的左结点进行RR旋转，再对根节点进行LL旋转
AvlNode<T> * q = RR(t->left);
t->left = q;
return LL(t);
}
//双旋转
//插入点位于t的右儿子的左子树
template <typename T>
AvlNode<T> * AvlTree<T>::RL(AvlNode<T> *t)
{
AvlNode<T> *q = LL(t->right);
t->right = q;
return RR(t);
}
template <typename T>
void AvlTree<T>::Insert(AvlNode<T> *&t, T x)
{
if (t == NULL)
t = new AvlNode<T>(x);
else if (x < t->data)
{
Insert(t->left, x);
//判断平衡情况
if (GetHeight(t->left) - GetHeight(t->right) > 1)
{
//分两种情况 左左或左右
if (x < t->left->data)//左左
t = LL(t);
else //左右
t = LR(t);
}
}
else if (x > t->data)
{
Insert(t->right, x);
if (GetHeight(t->right) - GetHeight(t->left) > 1)
{
if (x > t->right->data)
t = RR(t);
else
t = RL(t);
}
}
else
;//数据重复
t->height = max(GetHeight(t->left), GetHeight(t->right)) + 1;
}
template <typename T>
bool AvlTree<T>::Delete(AvlNode<T> *&t, T x)
{
//t为空 未找到要删除的结点
if (t == NULL)
return false;
//找到了要删除的结点
else if (t->data == x)
{
//左右子树都非空
if (t->left != NULL && t->right != NULL)
{//在高度更大的那个子树上进行删除操作
//左子树高度大，删除左子树中值最大的结点，将其赋给根结点
if (GetHeight(t->left) > GetHeight(t->right))
{
t->data = FindMax(t->left)->data;
Delete(t->left, t->data);
}
else//右子树高度更大，删除右子树中值最小的结点，将其赋给根结点
{
t->data = FindMin(t->right)->data;
Delete(t->right, t->data);
}
}
else
{//左右子树有一个不为空，直接用需要删除的结点的子结点替换即可
AvlNode<T> *old = t;
t = t->left ? t->left: t->right;//t赋值为不空的子结点
delete old;
}
}
else if (x < t->data)//要删除的结点在左子树上
{
//递归删除左子树上的结点
Delete(t->left, x);
//判断是否仍然满足平衡条件
if (GetHeight(t->right) - GetHeight(t->left) > 1)
{
if (GetHeight(t->right->left) > GetHeight(t->right->right))
{
//RL双旋转
t = RL(t);
}
else
{//RR单旋转
t = RR(t);
}
}
else//满足平衡条件 调整高度信息
{
t->height = max(GetHeight(t->left), GetHeight(t->right)) + 1;
}
}
else//要删除的结点在右子树上
{
//递归删除右子树结点
Delete(t->right, x);
//判断平衡情况
if (GetHeight(t->left) - GetHeight(t->right) > 1)
{
if (GetHeight(t->left->right) > GetHeight(t->left->left))
{
//LR双旋转
t = LR(t);
}
else
{
//LL单旋转
t = LL(t);
}
}
else//满足平衡性 调整高度
{
t->height = max(GetHeight(t->left), GetHeight(t->right)) + 1;
}
}
return true;
}
//查找结点
template <typename T>
bool AvlTree<T>::Contains(AvlNode<T> *t, const T x) const
{
if (t == NULL)
return false;
if (x < t->data)
return Contains(t->left, x);
else if (x > t->data)
return Contains(t->right, x);
else
return true;
}
//中序遍历
template <typename T>
void AvlTree<T>::InorderTraversal(AvlNode<T> *t)
{
if (t)
{
InorderTraversal(t->left);
cout << t->data << ‘ ‘;
InorderTraversal(t->right);
}
}
//前序遍历
template <typename T>
void AvlTree<T>::PreorderTraversal(AvlNode<T> *t)
{
if (t)
{
cout << t->data << ‘ ‘;
PreorderTraversal(t->left);
PreorderTraversal(t->right);
}
}
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