二叉树的实现与操作(C语言实现)

时间：2022-01-14 10:10:22
     

     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 1

typedef 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: 4
Degree: 2
Count: 6
Position At (0x02, 2): E
Full Tree:
A
----B
--------D
--------E
------------F
------------
----C
After Delete B:
Height: 2
Degree: 1
Count: 2
Full Tree:
A
----
----C
After Clear:
Height: 0
Degree: 0
Count: 0

请按任意键继续. . .



如有错误，望不吝指出呀。


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