算法导论 第十二章：二叉查找树(Binary Search Trees)

      二叉查找树具有如下性质：

       x是二叉查找树中的一个节点，如果y是x左子树中的一个节点，则y.key ≤ x.key ; 如果 y 是 x 右子树中的一个节点，则 x.key ≥ y.key.

        在二叉树上执行的基本操作的时间与树的高度成正比。当这棵树是完全二叉树时，这些操作的最坏情况运行时间为Θ(lgn);如果该树是含n个节点的线性链，则这些操作的最坏情况的运行时间为Θ(n)。我们可以通过随机构造二叉查找树(期望高度：E(h)=O(lgn))，从而使得在这种树上基本动态集操作的平均时间为Θ(lgn)。

基本操作：

查找关键字

最大值&最小值

查找某key的前驱后继

插入

删除

完整代码如下：

#include<iostream>
#include<cstdlib>
using namespace std;

typedef struct BSTNode{
int key; //key value
//int data; //satellite data
BSTNode *parent; //parent node
BSTNode *left; //left child node
BSTNode *right; //right child node
}BSTNode;

typedef struct BSTree
{
BSTNode *root;
}BSTree;

void BST_Insert(BSTree *T,int value)
{
BSTNode *x=T->root;
BSTNode *y=NULL; //y is x's parent node

BSTNode *z=new BSTNode();//new inserted node
z->key=value;
z->parent=NULL;
z->left=NULL;
z->right=NULL;

//locate insert position
while(x!=NULL)
{
y=x;
if(z->key < x->key)
x=x->left;
else
x=x->right;
}
//link new node to its parent node
z->parent = y;

//link parent node to new node
if(y==NULL) //Treee is empty
T->root=z;
else if(z->key < y->key)
y->left = z;
else
y->right = z;
}
void BST_InorderWalk(BSTNode *x)
{
if(x!=NULL)
{
BST_InorderWalk(x->left);
cout<<x->key<<" ";
BST_InorderWalk(x->right);
}
}
BSTNode *BST_Search(BSTNode *x,int skey)
{
if(x==NULL || skey==x->key)
return x;
if(skey < x->key)
return BST_Search(x->left,skey);
else
return BST_Search(x->right,skey);
}
BSTNode *BST_Minimum(BSTNode *x)
{
while(x->left!=NULL)
x=x->left;
return x;
}
BSTNode *BST_Maximum(BSTNode *x)
{
while(x->right!=NULL)
x=x->right;
return x;
}
BSTNode *BST_Successor(BSTNode *x)
{
if(x->right!=NULL)
return BST_Minimum(x->right);
BSTNode *y=x->parent;
while(y!=NULL && x==y->right)
{
x=y;
y=y->parent;
}
return y;
}
BSTNode *BST_Predecessor(BSTNode *x)
{
if(x->left !=NULL)
return BST_Maximum(x->left);
BSTNode *y=x->parent;
while(y!=NULL && x==y->left)
{
x=y;
y=y->parent;
}
return y;
}
void BST_Transplant(BSTree *T,BSTNode *u,BSTNode *v)
{
if(u->parent==NULL)
T->root=v;
else if(u==u->parent->left)
u->parent->left=v;
else
u->parent->right=v;
if(v!=NULL)
v->parent=u->parent;
}
void BST_Delete(BSTree *T,BSTNode *z)
{
if(z->left==NULL) //case 1
BST_Transplant(T,z,z->right);
else if(z->right==NULL) //case 2
BST_Transplant(T,z,z->left);
else //case 3
{
BSTNode *y=BST_Minimum(z->right);
if(y->parent!=z)
{
BST_Transplant(T,y,y->right);
y->right=z->right;
y->right->parent=y;
}
BST_Transplant(T,z,y);
y->left=z->left;
y->left->parent=y;
}
}
int main()
{
int A[]={12,5,2,9,18,15,17,19};
int n=sizeof(A)/sizeof(int);

cout<<"/*----------------------Create BST-------------------------*/"<<endl;
BSTree *T=new BSTree();
T->root =NULL;
for(int i=0;i<n;i++) //Create Binary Search Tree
BST_Insert(T,A[i]);

cout<<"The Tree is(Inorder-walk-tree):"<<endl;
BST_InorderWalk(T->root);
cout<<endl;
cout<<"/*---------------------------------------------------------*/"<<endl;

cout<<"/*-----------------------Search BST------------------------*/"<<endl;
int skey;
BSTNode *snode=NULL;

while(1)
{
cout<<"Please input the searching key(-1 denote exiting):";
cin>>skey;
if(skey==-1)
{
cout<<"Exiting..."<<endl;
break;
}
snode=BST_Search(T->root,skey);
if(snode!=NULL)
cout<<"The node exists in the tee!"<<endl;
else
cout<<"The node doesn't exist."<<endl;
}
cout<<"/*----------------------------------------------------------*/"<<endl;

cout<<"/*-------------------Minimum and Maximum--------------------*/"<<endl;
BSTNode *minNode=NULL,*maxNode=NULL;
minNode=BST_Minimum(T->root);
maxNode=BST_Maximum(T->root);
cout<<"The Minimum of the BST is:"<<minNode->key<<endl;
cout<<"The Maximum of the BST is:"<<maxNode->key<<endl;
cout<<"/*----------------------------------------------------------*/"<<endl;

cout<<"/*-------------------Successor and Predecessor--------------*/"<<endl;
cout<<"Please input the node's key:";
cin>>skey;
BSTNode *curNode=NULL;
curNode=BST_Search(T->root,skey);
if(curNode!=NULL)
{
BSTNode *sucNode=NULL,*preNode=NULL;
sucNode=BST_Successor(curNode);
preNode=BST_Predecessor(curNode);
cout<<"The Successor of the current Node "<<curNode->key<<" is:"<< sucNode->key<<endl;
cout<<"The Predecessor of the current Node "<<curNode->key<<" is:"<< preNode->key<<endl;
}
else
cout<<"The node doen't exist."<<endl;
cout<<"/*-----------------------------------------------------------*/"<<endl;

cout<<"/*----------------------Insert-------------------------------*/"<<endl;
int ikey;
cout<<"Please input the iserting node's key:";
cin>>ikey;
BST_Insert(T,ikey);
cout<<"After inserting ,the Tree is:"<<endl;
BST_InorderWalk(T->root);
cout<<endl;
cout<<"/*-----------------------------------------------------------*/"<<endl;

cout<<"/*------------------Deletion---------------------------------*/"<<endl;
int dkey;
cout<<"Please input the deleting node's key:";
cin>>dkey;
BSTNode *dNode=NULL;
dNode=BST_Search(T->root,dkey);
if(dNode==NULL)
cout<<"The node doesn't exist in the treee."<<endl;
else
{
BST_Delete(T,dNode);
cout<<"After deleting ,the tree is:"<<endl;
BST_InorderWalk(T->root);
cout<<endl;
}
cout<<"/*-----------------------------------------------------------*/"<<endl;

return 0;
}
运行结果：