二叉树的实现与操作(C语言实现)

     

     二叉树的定义：

    上一篇的树的通用表示法太过于复杂，由此这里采用了孩子兄弟表示法来构建二叉树。

孩子兄弟表示法:

   每个结点包含一个数据指针和两个结点指针

--->数据指针：指向保存于树中的数据

--->孩子结点指针：指向第一个孩子

--->兄弟结点指针：指向第一个右兄弟

二叉树是由 n( n>=0 ) 个结点组成的有限集合,该集合或者为空,或者是由一个根结点加上两棵分别称为左子树和右子树的、互不相交的二叉树组成。

特殊的二叉树：

定义1：满二叉树(Full Binary Tree)

如果二叉树中所有分支结点的度数都为2，且叶子结点都在同一层次上，则称做这类二叉树为满二叉树

定义2：完全二叉树

如果一颗具有N个结点的高度为K的二叉树，它的每一个结点都与高度为K的满二叉树中的编号为1---N的结点一一对应，则称这课二叉树为完全二叉树（从上到下从左到右编号）。

注：完全二叉树的叶结点仅仅出现在最下面二层，

      最下面的叶结点一定出现在左边；

      倒数第二层的叶结点一定出现在右边

  完全二叉树中度为1的结点只有左孩子

  同样结点数的二叉树，完全二叉树的高度最小

二叉树的深层性质

 

性质1：

 在二叉树的第i层最多有2i-1个结点。（i>=1）

性质2：

 深度为K的二叉树最多有2k-1个结点（k>=0）

性质3：

 对任何一颗二叉树，如果其叶结点有n0个，度为2的结点的非叶结点有n2个，则有n0=n2+1

性质4：

 具有n个结点的完全二叉树的高度为[log2n]+1

 

性质5：

  一颗有n个结点的二叉树（高度为[log2n]+1），按层次对结点进行编号（从上到下，从左到右），对任意结点i有：

   如果i=1,则结点i是二叉树的根，

   如果i>1,则其双亲结点为[i/2]

   如果2i<=n,则结点i的左孩子为2i

   如果2i>n,则结点i无左孩子，

   如果2i+1<=n,则结点i的右孩子为2i+1,

   如果2i+1>n,则结点i无右孩子

以下是代码：

头文件：

#ifndef _BTREE_H_#define _BTREE_H_#define BT_LEFT 0#define BT_RIGHT 1typedef void BTree;     //树typedef unsigned long long BTPos;  //要插入结点的位置，是一个十六进制数字typedef struct _tag_BTreeNode BTreeNode;   //定义树结点struct _tag_BTreeNode{BTreeNode* left;BTreeNode* right;};typedef void (BTree_Printf)(BTreeNode*);BTree* BTree_Create();void BTree_Destroy(BTree* tree);void BTree_Clear(BTree* tree);int BTree_Insert(BTree* tree, BTreeNode* node, BTPos pos, int count, int flag);BTreeNode* BTree_Delete(BTree* tree, BTPos pos, int count);BTreeNode* BTree_Get(BTree* tree, BTPos pos, int count);BTreeNode* BTree_Root(BTree* tree);int BTree_Height(BTree* tree);int BTree_Count(BTree* tree);int BTree_Degree(BTree* tree);void BTree_Display(BTree* tree, BTree_Printf* pFunc, int gap, char div);#endif

源文件：

#include "stdafx.h"#include <stdio.h>#include <malloc.h>#include "BTree.h"typedef struct _tag_BTree TBTree;struct _tag_BTree    //树的头结点定义{int count;BTreeNode* root;};//打印函数static void recursive_display(BTreeNode* node, BTree_Printf* pFunc, int format, int gap, char div) {int i = 0;if( (node != NULL) && (pFunc != NULL) ){//先打印格式符号for(i=0; i<format; i++){printf("%c", div);}//打印树中具体的数据pFunc(node);printf("\n");//如果左 或者 右结点不为空才打印if( (node->left != NULL) || (node->right != NULL) ){recursive_display(node->left, pFunc, format + gap, gap, div);recursive_display(node->right, pFunc, format + gap, gap, div);}}//如果结点为空 就打印 格式符号else{for(i=0; i<format; i++){printf("%c", div);}printf("\n");}}//统计树中结点的数量static int recursive_count(BTreeNode* root) {int ret = 0;if( root != NULL ){ret = recursive_count(root->left) + 1 + recursive_count(root->right);}return ret;}//计算树的高度static int recursive_height(BTreeNode* root) {int ret = 0;if( root != NULL ){int lh = recursive_height(root->left);int rh = recursive_height(root->right);ret = ((lh > rh) ? lh : rh) + 1;}return ret;}//计算树的度static int recursive_degree(BTreeNode* root) {int ret = 0;if( root != NULL ){if( root->left != NULL ){ret++;}if( root->right != NULL ){ret++;}if( ret == 1 ){int ld = recursive_degree(root->left);int rd = recursive_degree(root->right);if( ret < ld ){ret = ld;}if( ret < rd ){ret = rd;}}}return ret;}BTree* BTree_Create() {TBTree* ret = (TBTree*)malloc(sizeof(TBTree));if( ret != NULL ){ret->count = 0;ret->root = NULL;}return ret;}void BTree_Destroy(BTree* tree){free(tree);}void BTree_Clear(BTree* tree) {TBTree* btree = (TBTree*)tree;if( btree != NULL ){btree->count = 0;btree->root = NULL;}}//tree  目标树  node 要插入结点   pos 要插入位置  count 移动步数  flag  插入位置是左还是右int BTree_Insert(BTree* tree, BTreeNode* node, BTPos pos, int count, int flag) {TBTree* btree = (TBTree*)tree;int ret = (btree != NULL) && (node != NULL) && ((flag == BT_LEFT) || (flag == BT_RIGHT));int bit = 0;if( ret ){BTreeNode* parent = NULL;BTreeNode* current = btree->root;node->left = NULL;node->right = NULL;while( (count > 0) && (current != NULL) ){//位置最低位与1进行按位与运算，得知是往左走还是往右走bit = pos & 1;//表示位置的十六进制向右移动一位pos = pos >> 1;//parent用来挂要插入的结点parent = current;if( bit == BT_LEFT ){current = current->left;}else if( bit == BT_RIGHT ){current = current->right;}count--;}//插入的结点挂上中间被砍断的剩下的结点if( flag == BT_LEFT ){node->left = current;}else if( flag == BT_RIGHT ){node->right = current;}//将要插入的结点挂上if( parent != NULL ){if( bit == BT_LEFT ){parent->left = node;}else if( bit == BT_RIGHT ){parent->right = node;}}else{btree->root = node;}btree->count++;}return ret;}//删除与插入基本类似，只不过将要删除的结点的父结点的left或者right指针以及所有的子节点置为NULL而已BTreeNode* BTree_Delete(BTree* tree, BTPos pos, int count) {TBTree* btree = (TBTree*)tree;BTreeNode* ret = NULL; int bit = 0;if( btree != NULL ){BTreeNode* parent = NULL;BTreeNode* current = btree->root;while( (count > 0) && (current != NULL) ){bit = pos & 1;pos = pos >> 1;parent = current;if( bit == BT_LEFT ){current = current->left;}else if( bit == BT_RIGHT ){current = current->right;}count--;}if( parent != NULL ){if( bit == BT_LEFT ){parent->left = NULL;}else if( bit == BT_RIGHT ){parent->right = NULL;}}else{btree->root = NULL;}ret = current;btree->count = btree->count - recursive_count(ret);}return ret;}BTreeNode* BTree_Get(BTree* tree, BTPos pos, int count) {TBTree* btree = (TBTree*)tree;BTreeNode* ret = NULL; int bit = 0;if( btree != NULL ){BTreeNode* current = btree->root;while( (count > 0) && (current != NULL) ){bit = pos & 1;pos = pos >> 1;if( bit == BT_LEFT ){current = current->left;}else if( bit == BT_RIGHT ){current = current->right;}count--;}ret = current;}return ret;}BTreeNode* BTree_Root(BTree* tree){TBTree* btree = (TBTree*)tree;BTreeNode* ret = NULL;if( btree != NULL ){ret = btree->root;}return ret;}int BTree_Height(BTree* tree) {TBTree* btree = (TBTree*)tree;int ret = 0;if( btree != NULL ){ret = recursive_height(btree->root);}return ret;}int BTree_Count(BTree* tree) {TBTree* btree = (TBTree*)tree;int ret = 0;if( btree != NULL ){ret = btree->count;}return ret;}int BTree_Degree(BTree* tree) {TBTree* btree = (TBTree*)tree;int ret = 0;if( btree != NULL ){ret = recursive_degree(btree->root);}return ret;}void BTree_Display(BTree* tree, BTree_Printf* pFunc, int gap, char div) {TBTree* btree = (TBTree*)tree;if( btree != NULL ){recursive_display(btree->root, pFunc, 0, gap, div);}}

主函数：

// 二叉树.cpp : 定义控制台应用程序的入口点。//#include "stdafx.h"#include "BTree.h"#include <iostream>struct Node    //数据结点{BTreeNode header;char v;};void printf_data(BTreeNode* node)   //打印树{if( node != NULL ){printf("%c", ((struct Node*)node)->v);}}int _tmain(int argc, _TCHAR* argv[]){BTree* tree = BTree_Create();struct Node n1 = {{NULL, NULL}, 'A'};struct Node n2 = {{NULL, NULL}, 'B'};struct Node n3 = {{NULL, NULL}, 'C'};struct Node n4 = {{NULL, NULL}, 'D'};struct Node n5 = {{NULL, NULL}, 'E'};struct Node n6 = {{NULL, NULL}, 'F'};BTree_Insert(tree, (BTreeNode*)&n1, 0, 0, 0);BTree_Insert(tree, (BTreeNode*)&n2, 0x00, 1, 0);BTree_Insert(tree, (BTreeNode*)&n3, 0x01, 1, 0);BTree_Insert(tree, (BTreeNode*)&n4, 0x00, 2, 0);BTree_Insert(tree, (BTreeNode*)&n5, 0x02, 2, 0);BTree_Insert(tree, (BTreeNode*)&n6, 0x02, 3, 0);printf("Height: %d\n", BTree_Height(tree));printf("Degree: %d\n", BTree_Degree(tree));printf("Count: %d\n", BTree_Count(tree));printf("Position At (0x02, 2): %c\n", ((struct Node*)BTree_Get(tree, 0x02, 2))->v);printf("Full Tree: \n");BTree_Display(tree, printf_data, 4, '-');//以下是删除结点位置在0x00的结点后，树的整体状态BTree_Delete(tree, 0x00, 1);printf("After Delete B: \n");printf("Height: %d\n", BTree_Height(tree));printf("Degree: %d\n", BTree_Degree(tree));printf("Count: %d\n", BTree_Count(tree));printf("Full Tree: \n");BTree_Display(tree, printf_data, 4, '-');//以下是清空树后，树的整体状态BTree_Clear(tree);printf("After Clear: \n");printf("Height: %d\n", BTree_Height(tree));printf("Degree: %d\n", BTree_Degree(tree));printf("Count: %d\n", BTree_Count(tree));BTree_Display(tree, printf_data, 4, '-');BTree_Destroy(tree);system("pause");return 0;}

运行结构：

Height: 4Degree: 2Count: 6Position At (0x02, 2): EFull Tree:A----B--------D--------E------------F----------------CAfter Delete B:Height: 2Degree: 1Count: 2Full Tree:A--------CAfter Clear:Height: 0Degree: 0Count: 0请按任意键继续. . .

如有错误，望不吝指出呀。