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 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。

二叉树的实现与操作(C语言实现)