一步一步学数据结构之1--n（二叉树）

 

        既然树已经熟悉了，那我们就来学习学习二叉树吧，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。

        如图

        

有两个定义需要大家知道下：

1.满二叉树

        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，则称这类二叉树为满二叉树。

2.完全二叉树

        如果一棵具有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的满二叉树中编号为1-n的结点一一对应，则称这棵二叉树为完全二叉树。（从上到下从左到右编号）

        完全二叉树的叶结点仅出现在最下面两层

        最下层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

二叉树所具有的5个性质需要大家掌握：

 

这里介绍通用树的常用操作：

l 创建二叉树

l 销毁二叉树

l 清空二叉树

l 插入结点到二叉树中

l 删除结点

l 获取某个结点

l 获取根结点

l 获取二叉树的高度

l 获取二叉树的总结点数

l 获取二叉树的度

l 输出二叉树

代码总分为三个文件：

BTree.h ： 放置功能函数的声明，以及树的声明，以及树结点的定义 

BTree.c ： 放置功能函数的定义，以及树的定义

Main.c   ： 主函数，使用功能函数完成各种需求，一般用作测试

整体结构图为：

这里详细说下插入结点操作，删除结点操作和获取结点操作：

 

插入结点操作：

如图：

  

删除结点操作：

如图：

获取结点操作：

            获取结点操作和插入删除结点操作中的指路法定位结点相同

OK! 上代码：

BTree.h ： 

#ifndef _BTREE_H_#define _BTREE_H_#define BT_LEFT 0#define BT_RIGHT 1typedef void BTree;typedef unsigned long long BTPos;typedef struct _tag_BTreeNode BTreeNode;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

 

BTree.c ： 

#include <stdio.h>#include <malloc.h>#include "BTree.h"typedef struct _tag_BTree TBTree;struct _tag_BTree{int count;BTreeNode* root;};BTree* BTree_Create(){TBTree* ret = (TBTree*)malloc(sizeof(TBTree));if(NULL != ret){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(NULL != btree){btree->count = 0;btree->root = NULL;}}int BTree_Insert(BTree* tree, BTreeNode* node, BTPos pos, int count, int flag){TBTree* btree = (TBTree*)tree;int ret = (NULL!=btree) && (NULL!=node) && ((flag == BT_RIGHT) || (flag == BT_LEFT));int bit = 0;if(ret){BTreeNode* parent = NULL;BTreeNode* current = btree->root;node->left = NULL;node->right = NULL;while((0 < count) && (NULL != current)){bit = pos & 1;pos = pos >> 1;parent = current;if(BT_LEFT == bit){current = current->left;}else if(BT_RIGHT == bit){current = current->right;}count--;}if(BT_LEFT == flag){node->left = current;}else if(BT_RIGHT == flag){node->right = current;}if(NULL != parent){if(BT_LEFT == bit){parent->left = node;}else if(BT_RIGHT == bit){parent->right = node;}}else{btree->root = node;}btree->count++;}return ret;}static int recursive_count(BTreeNode* root){int ret = 0;if(NULL != root){ret = recursive_count(root->left) + 1 +   recursive_count(root->right);}return ret;}BTreeNode* BTree_Delete(BTree* tree, BTPos pos, int count){TBTree* btree = (TBTree*)tree;BTreeNode* ret = NULL;int bit = 0;if(NULL != btree){BTreeNode* parent = NULL;BTreeNode* current = btree->root;while((0 < count) && (NULL != current)){bit = pos & 1;pos = pos >> 1;parent = current;if(BT_RIGHT == bit){current = current->right;}else if(BT_LEFT == bit){current = current->left;}count--;}if(NULL != parent){if(BT_LEFT == bit){parent->left = NULL;}else if (BT_RIGHT == bit){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(NULL != btree){BTreeNode* current = btree->root;while((0<count) && (NULL!=current)){bit = pos & 1;pos = pos >> 1;if(BT_RIGHT == bit){current = current->right;}else if(BT_LEFT == bit){current = current->left;}count--;}ret = current;}return ret;}BTreeNode* BTree_Root(BTree* tree){TBTree* btree = (TBTree*)tree;BTreeNode* ret = NULL;if(NULL != btree){ret = btree->root;}return ret;}static int recursive_height(BTreeNode* root){int ret = 0;if(NULL != root){int lh = recursive_height(root->left);int rh = recursive_height(root->right);ret = ((lh > rh) ? lh : rh) + 1;}return ret;}int BTree_Height(BTree* tree){TBTree* btree = (TBTree*)tree;int ret = -1;if(NULL != btree){ret = recursive_height(btree->root);}return ret;}int BTree_Count(BTree* tree){TBTree* btree = (TBTree*)tree;int ret = -1;if(NULL != btree){ret = btree->count;}return ret;}static int recursive_degree(BTreeNode* root){int ret = 0;if(NULL != root){if(NULL != root->left){ret++;}if(NULL != root->right){ret++;}if(1 == ret){int ld = recursive_degree(root->left);int rd = recursive_degree(root->right);if(ret < ld){ret = ld;}if(ret < rd){ret = rd;}}}return ret;}int BTree_Degree(BTree* tree){TBTree* btree = (TBTree*)tree;int ret = -1;if(NULL != btree){ret = recursive_degree(btree->root);}return ret;}static void recursive_display(BTreeNode* node, BTree_Printf* pFunc, int format, int gap, char div){int i = 0;if((NULL != node) && (NULL != pFunc)){for(i=0; i<format; i++){printf("%c", div);}pFunc(node);printf("\n");if((NULL != node->left) || (NULL != node->right)){recursive_display(node->left, pFunc, format+gap, gap, div);recursive_display(node->right, pFunc, format+gap, gap, div);}}else{for(i=0; i<format; i++){printf("%c", div);}printf("\n");}}void BTree_Display(BTree* tree, BTree_Printf* pFunc, int gap, char div){TBTree* btree = (TBTree*)tree;if(NULL != btree){recursive_display(btree->root, pFunc, 0, gap, div);}}

 

Main.c  ：

#include <stdio.h>#include <stdlib.h>#include "BTree.h"typedef struct _tag_node{BTreeNode header;char v;}Node;void printf_data(BTreeNode* node){if(NULL != node){printf("%c", ((Node*)node)->v);}}int main(void){BTree* tree = BTree_Create();Node n1 = {{NULL, NULL}, 'A'};Node n2 = {{NULL, NULL}, 'B'};Node n3 = {{NULL, NULL}, 'C'};Node n4 = {{NULL, NULL}, 'D'};Node n5 = {{NULL, NULL}, 'E'};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", ((Node*)BTree_Get(tree, 0x02, 2))->v);printf("Full Tree:\n");BTree_Display(tree, printf_data, 4, '-');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, '-');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));printf("Full Tree:\n");BTree_Display(tree, printf_data, 4, '-');BTree_Destroy(tree);return 0;}