您的位置:首页 > 其它

二叉查找树(binary search tree (BST))--算法导论示例

2010-05-22 23:56 721 查看
二叉查找树(Binary Search Tree),或者是一棵空树,或者是具有下列性质的二叉树:

1. 若它的左子树不空,则左子树上所有结点的值均小于它的根结点的值;
2. 若它的右子树不空,则右子树上所有结点的值均大于它的根结点的值;
3. 它的左、右子树也分别为二叉排序树。

二叉排序树的查找过程和次优二叉树类似,通常采取二叉链表作为二叉排序树的存储结构。中序遍历二叉排序树可得到一个关键字的有序序列,一个无序序列可以通过构造一棵二叉排序树变成一个有序序列,构造树的过程即为对无序序列进行排序的过程。每次插入的新的结点都是二叉排序树上新的叶子结点,在进行插入操作时,不必移动其它结点,只需改动某个结点的指针,由空变为非空即可。搜索,插入,删除的复杂度等于树高,O(log(n)).

二叉排序树的查找算法
在二叉排序树b中查找x的过程为:

若b是空树,则搜索失败,否则:
若x等于b的根结点的数据域之值,则查找成功;否则:
若x小于b的根结点的数据域之值,则搜索左子树;否则:
查找右子树。
在二叉排序树插入结点的算法
向一个二叉排序树b中插入一个结点s的算法,过程为:

若b是空树,则将s所指结点作为根结点插入,否则:
若s->data等于b的根结点的数据域之值,则返回,否则:
若s->data小于b的根结点的数据域之值,则把s所指结点插入到左子树中,否则:
把s所指结点插入到右子树中。
在二叉排序树删除结点的算法
在二叉排序树删去一个结点,分三种情况讨论:

若*p结点为叶子结点,即PL(左子树)和PR(右子树)均为空树。由于删去叶子结点不破坏整棵树的结构,则只需修改其双亲结点的指针即可。
右*p结点只有左子树PL或右子树PR,此时只要令PL或PR直接成为其双亲结点*f的左子树即可,作此修改也不破坏二叉排序树的特性。
若*p结点的左子树和右子树均不空。在删去*p之后,为保持其它元素之间的相对位置不变,可按中序遍历保持有序进行调整,可以有两种做法:其一是令*p的左子树为*f的左子树,*s为*f左子树的最右下的结点,而*p的右子树为*s的右子树;其二是令*p的直接前驱(或直接后继)替代*p,然后再从二叉排序树中删去它的直接前驱(或直接后继)。
二叉排序树性能分析
每个结点的Ci为该结点的层次数。最坏情况下,当先后插入的关键字有序时,构成的二叉排序树蜕变为单支树,树的深度为n,其平均查找长度为(n+1)/2(和顺序查找相同),最好的情况是二叉排序树的形态和折半查找的判定树相同,其平均查找长度和log2n成正比(O(log2n))。

#include "stdio.h"
#include "stdlib.h"

struct Node
...{
int key;//结点值
struct Node *parent;//父结点
struct Node *left;//左结点
struct Node *right;//右结点
};

//打印二叉树
void PrintTree(struct Node *Root)
...{
struct Node *pointer=(struct Node*)malloc(sizeof(struct Node));
pointer=Root;
if(pointer!=NULL)
...{
PrintTree(pointer->left);
printf("%d ",pointer->key);
PrintTree(pointer->right);
}
}

//删除树,递归释放树的结点
void DeleteTree(struct Node *Root)
...{
struct Node *pointer=(struct Node*)malloc(sizeof(struct Node));
pointer=Root;

if(pointer!=NULL)
...{
DeleteTree(pointer->left);
free(pointer);
DeleteTree(pointer->right);
}
}

//最小值
struct Node *Minimum(struct Node *Root)
...{
struct Node *pointer=(struct Node*)malloc(sizeof(struct Node));
pointer=Root;

while(pointer->left!=NULL)
pointer=pointer->left;

return pointer;
}

//最大值
struct Node *Maximum(struct Node *Root)
...{
struct Node *pointer=(struct Node*)malloc(sizeof(struct Node));
pointer=Root;

while(pointer->right!=NULL)
pointer=pointer->right;

return pointer;
}

//查找
struct Node *Search(struct Node *Root,int value)
...{
struct Node *pointer=(struct Node*)malloc(sizeof(struct Node));
pointer=Root;

while(pointer!=NULL&&pointer->key!=value)
...{
//如果查找值比结点值小,查找它的左子树
if(value<pointer->key)
pointer=pointer->left;
else
pointer=pointer->right;//如果查找值比结点值大,查找它的右子树
}

return pointer;
}

//后继
struct Node *Successor(struct Node *pointer)
...{
if(pointer->right!=NULL)
return Minimum(pointer->right);

struct Node *y=(struct Node*)malloc(sizeof(struct Node));
y=pointer->parent;
while(y!=NULL&&y->right==pointer)
...{
pointer=y;
y=pointer->parent;
}

return y;
}

//前驱
struct Node *Predecessor(struct Node *pointer)
...{
if(pointer->left!=NULL)
return Maximum(pointer->left);

struct Node *y=(struct Node*)malloc(sizeof(struct Node));
y=pointer->parent;
while(y!=NULL&&y->left==pointer)
...{
pointer=y;
y=pointer->parent;
}

return y;
}

//插入新结点
struct Node *Insert(struct Node *Root,struct Node *newNode)
...{
struct Node *back=(struct Node*)malloc(sizeof(struct Node));
struct Node *pointer=(struct Node*)malloc(sizeof(struct Node));
back=NULL;
pointer=Root;

//找到要插入的位置
while(pointer!=NULL)
...{
back=pointer;
if(newNode->key<pointer->key)
pointer=pointer->left;
else
pointer=pointer->right;
}

//新结点的Parent指针先指向父结点
newNode->parent=back;

if(back==NULL)
Root=newNode;//树为空,新结点成为树根
else
...{
if(newNode->key<back->key)
back->left=newNode;
else
back->right=newNode;
}

return Root;
}

//删除结点
struct Node *Delete(struct Node *Root,struct Node *pointer)
...{
struct Node *y=(struct Node*)malloc(sizeof(struct Node));
y->parent=NULL;
y->left=NULL;
y->right=NULL;
//找到实际要删除的结点
if(pointer->left==NULL||pointer->right==NULL)
y=pointer;
else
y=Successor(pointer);

struct Node *x=(struct Node*)malloc(sizeof(struct Node));
x->parent=NULL;
x->left=NULL;
x->right=NULL;
//要删除结点的子结点
if(y->left!=NULL)
x=y->left;
else
x=y->right;

if(x!=NULL)
x->parent=y->parent;

if(y->parent==NULL)
Root=x;//删除的是根结点
else
...{
if(y==y->parent->left)
y->parent->left=x;
else
y->parent->right=x;
}

if(y!=pointer)
pointer->key=y->key;

if(y!=NULL)
...{
free(y);
printf("Delete success! ");
}
else
printf("Delete failure! ");

return Root;
}

int main()
...{
struct Node *Root=NULL;
int value;

int flag=1;
while(flag)
...{
printf("1:Insert ");
printf("2:Delete ");
printf("3:Search ");
printf("4:Minimum ");
printf("5:Maximum ");
printf("6:Successor ");
printf("7:Predecessor ");
printf("0:Exit ");
printf("Please input your choice:");
scanf("%d",&flag);

switch(flag)
...{
case 0:
return 0;
case 1:
...{
printf("Please input the value you want to input:");
scanf("%d",&value);
struct Node *newNode=(struct Node*)malloc(sizeof(struct Node));
newNode->key=value;
newNode->parent=NULL;
newNode->left=NULL;
newNode->right=NULL;
Root=Insert(Root,newNode);
break;
}
case 2:
...{
printf("Please input the value you want to delete:");
scanf("%d",&value);
struct Node *newNode=(struct Node*)malloc(sizeof(struct Node));
newNode=Search(Root,value);
if(newNode!=NULL)
Root=Delete(Root,newNode);
else
printf("%d does'n exist in this tree! ",value);
break;
}
case 3:
...{
printf("Please input the value you want to input:");
scanf("%d",&value);
struct Node *newNode=(struct Node*)malloc(sizeof(struct Node));
newNode=Search(Root,value);
if(newNode!=NULL)
printf("%d exist in this tree! ",newNode->key);
else
printf("%d does'n exist in this tree! ",value);
break;
}
case 4:
...{
if(Root!=NULL)
...{
struct Node *newNode=(struct Node*)malloc(sizeof(struct Node));
newNode=Minimum(Root);
printf("The minimum value of this tree is %d! ",newNode->key);
}
else
printf("This tree is empty! ");
break;
}
case 5:
...{
if(Root!=NULL)
...{
struct Node *newNode=(struct Node*)malloc(sizeof(struct Node));
newNode=Maximum(Root);
printf("The Maximum value of this tree is %d! ",newNode->key);
}
else
printf("This tree is empty! ");
break;
}
case 6:
...{
printf("Please input the value:");
scanf("%d",&value);
struct Node *newNode=(struct Node*)malloc(sizeof(struct Node));
newNode=Search(Root,value);
if(newNode!=NULL)
...{
newNode=Successor(newNode);
if(newNode!=NULL)
printf("%d's successor is %d! ",value,newNode->key);
else
printf("%d has't successor int this tree! ",value);
}
else
printf("%d does'n exist in this tree! ",value);
break;
}
case 7:
...{
printf("Please input the value:");
scanf("%d",&value);
struct Node *newNode=(struct Node*)malloc(sizeof(struct Node));
newNode=Search(Root,value);
if(newNode!=NULL)
...{
newNode=Predecessor(newNode);
if(newNode!=NULL)
printf("%d's Predecessor is %d! ",value,newNode->key);
else
printf("%d has't predecessor int this tree! ",value);
}
else
printf("%d does'n exist in this tree! ",value);
break;
}
default:
break;
}

PrintTree(Root);
}

DeleteTree(Root);
return 0;
}

本文来自CSDN博客,转载请标明出处:http://blog.csdn.net/hedongfu/archive/2006/09/12/1211007.aspx
内容来自用户分享和网络整理,不保证内容的准确性,如有侵权内容,可联系管理员处理 点击这里给我发消息
标签: 
相关文章推荐