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
原文地址:http://blog.csdn.net/wanglx_/article/details/40662583