二叉排序树（C与Python分别实现）

1.      什么是二叉排序树？

二叉排序树是一种特殊的二叉树，可以是一棵空树，也可以是具有下列性质的二叉树：

1.      若左子树不为空，那么左子树所有结点的值都小于它的根结点的值。

2.      若右子树不为空，那么右子树所有结点的值都大于它的根节点的值。

3.      它的左右子树也分别是二叉排序树。

二叉排序树又称二叉查找树，是一种动态查找表，所谓动态查找表是指除了查询检索操作以外，还可以进行插入、删除操作的结构；与之相对应的就是静态查找表，只能进行查询检索操作。

2.      二叉排序树的操作

对于二叉排序树的实现就是对查找、插入、删除操作的实现，因为要进行插入删除操作，所以要用链表结构来实现，操作介绍如下：

1.   查找key：从根结点出发，若key大于根结点则向根结点的右侧走，跟这个根结点的右结点比较，若key小于根结点则向根结点的左侧走，跟这个根结点做结点比较，该根结点的左右子树分别都是二叉排序树，所以重复上面的步骤，直到找到key，或者到NULL（叶子结点的子结点为空）。

2.   插入key：跟查找key步骤相同，找到key时返回，不进行插入，未找到时，在最后一个查找叶子结点下插入key值，若key大于该叶子结点的值则插入该叶子结点的右侧，作为该叶子结点的右子树，否则作为左子树。

3.   删除key：跟查找key步骤相同，若查找最终为NULL（未找到），则返回，若在结点p处找到key，则进行一下判断：

    a.      若p为叶子结点，先找到p的父节点q，若p为q的左子树，则将q的左子树指针至NULL，若p为q的右子树，则将q的右子树指针至NULL，删除p

    b.      若p的左子树不为空，右子树为空，则找到p的父节点q，若p是q的左子树，则将q的左子树指针指向p的左子树，若p是q的右子树，则将q的右子树指针指向p的左子树，删除p

    c.      若p的右子树不为空，左子树为空，则找到p的父节点q，若p是q的左子树，则将q的左子树指针指向p的右子树，若p是q的右子树，则将q的右子树指针指向p的右子树，删除p

    d.      若p的左右子树都不为空，则找到p的父节点q，将p的左子树作为p右子树的最左结点的子结点，然判断p是q的右子树还是q的左子树，若p是q的左子树，则将q的左子树指针指向p的右子树，若p是q的右子树，则将q的右子树指针指向p的右子树，删除p

3.      二叉排序树代码实现：

C代码

注意：下面的查找操作使用的是递归方式实现，既然是递归就有递归深度的问题，太深容易造成堆栈溢出，在我电脑上最多可以进行10357次递归；可用循环替代递归。

#include <stdio.h>
typedef struct  TNode
{
int nKey;
struct TNode* lchild;
struct TNode* rchild;
struct TNode* parent;
}TNode;
//++++++++++++++++++++++++++查找检索操作++++++++++++++++++++++++++/
bool SearchBST(TNode* BST,int key,TNode* &pSearch,TNode* &pParent)
{
if ( !BST )
{
pSearch = pParent;
return false;
}
if ( BST->nKey == key )
{
pSearch = BST;
pParent = pSearch->parent;
return true;
}
else if ( BST->nKey>key )
{
pParent = BST;
return SearchBST(BST->lchild,key,pSearch,pParent);
}
else
{
pParent = BST;
return SearchBST(BST->rchild,key,pSearch,pParent);
}
return false;
}
//++++++++++++++++++++++++++结点插入操作++++++++++++++++++++++++++/
bool InsertNode(TNode* &BST,int key,TNode* &pSearch)
{
TNode* pParent = NULL;
if ( SearchBST(BST,key,pSearch,pParent) )
return false;
TNode* pNode = new TNode;
pNode->nKey = key;
pNode->parent = pParent;
pNode->lchild = NULL;
pNode->rchild = NULL;
if ( !pParent )
BST = pNode;
else if ( key>pParent->nKey )
pParent->rchild = pNode;
else
pParent->lchild = pNode;
pSearch = pNode;
return true;
}
//++++++++++++++++++++++++++删除结点操作++++++++++++++++++++++++++/
bool DeleteNode(TNode* BST,int key)
{
TNode* pSearch = NULL;
TNode* pParent = NULL;
if ( !SearchBST(BST,key,pSearch,pParent) )
return false;
if ( NULL!=pSearch->lchild && NULL!=pSearch->rchild )
{
TNode *pTmp = pSearch->rchild;
while(pTmp->lchild) pTmp = pTmp->lchild;
pTmp->lchild = pSearch->lchild;
pSearch->lchild->parent = pTmp->lchild;
if ( pParent->lchild == pSearch )
pParent->lchild = pSearch->rchild;
else
pParent->rchild = pSearch->rchild;
pSearch->rchild->parent=pParent;
}
else if ( NULL!=pSearch->lchild )
{
if( pParent->lchild == pSearch )
pParent->lchild = pSearch->lchild;
else
pParent->rchild = pSearch->lchild;
}
else if ( NULL!=pSearch->rchild )
{
if ( pParent->lchild == pSearch )
pParent->lchild = pSearch->rchild;
else
pParent->rchild = pSearch->rchild;
}
else
{
if( pParent->lchild == pSearch )
pParent->lchild = NULL;
else
pParent->rchild = NULL;
}
delete pSearch;
pSearch = NULL;
return true;
}
//++++++++++++++++++++++++++释放内存操作++++++++++++++++++++++++++/
void DeleteBST(TNode* BST)
{
if( !BST )return;
if ( !BST->lchild && !BST->rchild )
{
if( !BST->parent && BST->parent->lchild == BST )
BST->parent->lchild = NULL;
else if( !BST->parent && BST->parent->rchild == BST )
BST->parent->rchild = NULL;
delete BST;
BST = NULL;
return;
}
DeleteBST(BST->lchild);
DeleteBST(BST->rchild);
}
//++++++++++++++++++++++++++main测试++++++++++++++++++++++++++/
int main(int argc, char* argv[])
{
TNode* BST = NULL;
TNode* pNode = NULL;
TNode* pParent = NULL;
int nCount = 10358;
for ( int i=0;i<nCount;i++ )
{
if( InsertNode(BST,i,pNode) )
printf("插入%d成功\n",i);
}
if ( SearchBST(BST,nCount-1,pNode,pParent) )
{
printf("查找成功\n");
}
DeleteNode(BST,1);
if ( !SearchBST(BST,1,pNode,pParent) )
{
printf("查找失败");
}
DeleteBST(BST);
getchar();
return 0;
}

Python代码

第一次学习Python，代码中肯定有许多不足之处，希望指正，暂时实现如下：

#BST.py
_Debug = False
#+++++++++++++++++++++struct realized by class+++++++++++++++++++++#
class TNode:
def __init__(self,key,left,right,parent):
self.key=key
self.lchild=left
self.rchild=right
self.parent=parent
class PTNode:
def __init__(self,TNode):
self.TNode=TNode
#+++++++++++++++++++++++++search operation+++++++++++++++++++++++++#
def SearchBST(BST,key,search,parent):
if None==BST or None==BST.key:
return False
if key==BST.key:
parent.TNode=BST.parent
search.TNode=BST
return True
elif key<BST.key:
parent.TNode=BST
return SearchBST(BST.lchild,key,search,parent)
else:
parent.TNode=BST
return SearchBST(BST.rchild,key,search,parent)
return False
#+++++++++++++++++++++++++Insert operation+++++++++++++++++++++++++#
def InsertNode(BST,key,search):
Pparent=PTNode(BST)
Psearch=PTNode(BST)
if _Debug==True:
import pdb
pdb.set_trace()
if SearchBST(BST,key,Psearch,Pparent):
return False
parent=Pparent.TNode
search=Psearch.TNode
Node=TNode(key,None,None,None)
if None==parent or None==parent.key:
BST=Node
return True
if key>parent.key:
parent.rchild=Node
else:
parent.lchild=Node
Node.parent=parent
search=Node
return True
#+++++++++++++++++++++++++Delete operation+++++++++++++++++++++++++#
def DeleteNode(BST,key):
Pparent=PTNode(BST)
Psearch=PTNode(BST)
if False==SearchBST(BST,key,Psearch,Pparent):
return False
parent=Pparent.TNode
search=Psearch.TNode
if None!=search.lchild and None!=search.rchild:
tmp=search
while(None!=tmp.lchild):
tmp=tmp.lchild
tmp.lchild=search.lchild
if None!=parent and search==parent.lchild:

parent.lchild=search.rchild
search.rchild.parent=parent
elif None!=parent and search==parent.rchild:
parent.rchild=search.rchild
search.rchild.parent=parent

elif None!=search.lchild:
if None!=parent and search==parent.lchild:
parent.lchild=search.lchild
search.lchild.parent=parent
elif None!=parent and search==parent.rchild:
parent.rchild=search.lchild
search.lchild.parent=parent
elif None!=search.rchild:
if None!=parent and search==parent.lchild:
parent.lchild=search.rchild
search.rchild.parent=parent
elif None!=parent and search==parent.rchild:
parent.rchild=search.rchild
search.rchild.parent=parent
else:
parent.lchild=None
parent.rchild=None
del search
search=None
return True
#+++++++++++++++++++++++++Release operation+++++++++++++++++++++++++#
def DeleteBST(BST):
if None!=BST:
if None==BST.lchild and None==BST.rchild:
tmp=BST.parent
if tmp!=None and tmp.lchild==BST:
tmp.lchild=None
elif tmp!=None and tmp.rchild==BST:
tmp.rhicld=None
del BST
BST=None
return
DeleteBST(BST.lchild)
DeleteBST(BST.rchild)

#+++++++++++++++++++++++++Print  operation+++++++++++++++++++++++++#
def PrintBST(BST):
if None==BST:
return
print BST.key
PrintBST(BST.lchild)
PrintBST(BST.rchild)
#++++++++++++++++++++++++++Test operation++++++++++++++++++++++++++#
if __name__=='__main__':
BST=TNode(0,None,None,None)
Pparent=PTNode(BST)
Psearch=PTNode(BST)
for i in range(1,4):
if False==InsertNode(BST,i,Psearch.TNode):
print 'Insert Number %d Failed.'%(i)
print 'print BST:'
PrintBST(BST)
if True==SearchBST(BST,1,Psearch,Pparent):
print 'Find 1 in BST Sucess.'
if True==DeleteNode(BST,1):
print 'Delete 1 Success.'
if False==SearchBST(BST,1,Psearch,Pparent):
print 'Find 1 in BST Failed.'

DeleteBST(BST)
print 'Done'

4.      二叉排序树的查找复杂度

最坏复杂度为O(n),最好为O(1)

参考：《数据结构（C语言版）》严蔚敏 吴为民