二叉树特性及详细例子

时间：2021-08-27 15:33:49

二叉树的性质

一般二叉树性质：

  1. 在非空二叉树的k层上，至多有2k个节点(k>=0)
  2. 高度为k的二叉树中,最多有2k+1-1个节点(k>=0)
  3. 对于任何一棵非空的二叉树,如果叶节点个数为n0，度数为2的节点个数为n2，则有: n0 = n2 + 1

完全二叉树性质:

  1. 具有n个节点的完全二叉树的高度k为[log2n]
  2. 对于具有n个节点的完全二叉树,如果按照从上(根节点)到下(叶节点)和从左到右的顺序对二叉树中的所有节点从0开始到n-1进行编号,则对于任意的下标为k的节点，有：
  • 如果k=0,则它是根节点，它没有父节点；如果k>0,则它的父节点的下标为[(i-1)/2];
  • 如果2k+1 <= n-1,则下标为k的节点的左子结点的下标为2k+1;否则,下标为k的节点没有左子结点.
  • 如果2k+2 <= n-1,则下标为k的节点的右子节点的下标为2k+2;否则,下标为k的节点没有右子节点

满二叉树性质:

在满二叉树中，叶节点的个数比分支节点的个数多1

扩充二叉树性质:

  1. 在扩充二叉树中，外部节点的个数比内部节点的个数多1
  2. 对任意扩充二叉树，外部路径长度E和内部路径长度I之间满足以下关系：E = I + 2n, 其中n是内部节点个数

 

二叉树的实现与周游

二叉树的顺序表示：

二叉树的顺序表示适用于完全二叉树，根据完全二叉树的第2性质，可以建立处节点之间的关系，对于一般的二叉树，可以将其扩充为完全二叉树，然后用一维数组顺序存储.

以下是程序代码:

bintree.h

二叉树特性及详细例子
  1 #include<stdio.h>
  2 #include<stdlib.h>
  3 #define USED 1
  4 #define NOTUSED 0
  5 typedef int Type;
  6 
  7 // 二叉树节点结构体
  8 typedef struct treenode
  9 {
 10     int nodeIndex;
 11     // 标志是否为外部节点，如果是则为0，否则为1
 12     int isIn;
 13     // 0表示这个节点不在二叉树中，1表示在树中
 14     int isUsed;
 15     Type element;
 16 }TreeNode;
 17 
 18 // 二叉树结构体
 19 typedef struct bintree
 20 {
 21     int MAXN;
 22     int n;
 23     TreeNode *nodelist;
 24 }*Tree;
 25 
 26 // 创建最大节点为m的空二叉树
 27 Tree createBinTree(int m)
 28 {
 29     int i;
 30     Tree tree = NULL;
 31     TreeNode *nodelist = NULL;
 32     tree = malloc(sizeof(struct bintree));
 33     if (NULL != tree)
 34     {
 35         nodelist = malloc(m * sizeof(TreeNode));
 36         if (NULL != nodelist)
 37         {
 38             tree->MAXN = m;
 39             tree->n = -1;
 40             for (i = 0; i < m; i++)
 41             {
 42                 nodelist[i].isUsed = NOTUSED;
 43                 nodelist[i].nodeIndex = i;
 44             }
 45             printf("\n");    
 46             tree->nodelist = nodelist;
 47         }
 48     }
 49     return tree;
 50 }
 51 
 52 // 在二叉树末尾加入元素
 53 void inTree(Tree tree, Type x, int isIn)
 54 {
 55     void replaceTree(Tree, Type, int, int);
 56     if (tree->n + 1 < tree->MAXN)
 57     {
 58         replaceTree(tree, x, tree->n + 1, isIn);
 59         tree->n = tree->n + 1;
 60     }
 61     return;
 62 }
 63 
 64 // 替换二叉树中指定下标的元素
 65 void replaceTree(Tree tree, Type x, int index, int isIn)
 66 {
 67     if (tree->n == -1 || (index >= 0 && index <= tree->n + 1))
 68     {
 69         tree->nodelist[index].element = x;
 70         tree->nodelist[index].isIn = isIn;
 71         tree->nodelist[index].isUsed = USED;
 72     }
 73     return;
 74 }
 75 
 76 // 得到指定下标的元素
 77 Type getNode(Tree tree, int index)
 78 {
 79     int re;
 80     if (index >= 0 && index <= tree->n)
 81         re = tree->nodelist[index].element;
 82     return re;
 83 }
 84 
 85 // 得到父节点的元素下标
 86 int parent(Tree tree, int child)
 87 {
 88     if (child < 0 || child > tree->n)
 89         return -1;
 90     else
 91         return (child - 1) / 2;
 92 }
 93 
 94 // 得到右子节点的元素下标
 95 int rightChild(Tree tree, int parent)
 96 {
 97     if (parent < 0 || parent > tree->n)
 98         return -1;
 99     else
100         return 2 * parent + 2;
101 }
102 
103 // 得到左子节点的元素下标
104 int leftChild(Tree tree, int parent)
105 {
106     if (parent < 0 || parent > tree->n)
107         return -1;
108     else
109         return 2 * parent + 1;
110 }
二叉树特性及详细例子

 

这里用的是非递归的周游方法，所以要用到栈，一下是自定义的栈.

stack.h

二叉树特性及详细例子
 1 #include "bintree.h"
 2 typedef TreeNode DataType;
 3 
 4 typedef struct mystack
 5 {
 6     //数组最大长度
 7     int MAXN;
 8     //指定栈顶位置
 9     int index;
10     DataType *ele;
11     
12 }*MyStack, Stack;
13 
14 
15 //创建一个空栈
16 MyStack createStack(int num)
17 {
18     MyStack stack = NULL;
19     stack = malloc(sizeof(Stack));
20     if(stack != NULL)
21     {
22         stack->ele = malloc(sizeof(DataType) * num);
23         if(stack ->ele != NULL)
24         {
25             stack ->MAXN = num;
26             stack ->index = -1;
27         }
28     }
29     return stack;
30 }
31 
32 //判断栈是否为空
33 int isEmptyStack(MyStack stack)
34 {
35     return stack ->index == -1;
36 }
37 
38 //元素入栈,如果栈已经满则返回0,否则返回1
39 int push(MyStack stack, DataType x)
40 {
41     int index = stack ->index + 1;
42     if(index >= stack ->MAXN)
43         return 0;
44     else
45     {
46         stack ->ele[index] = x;
47         stack ->index = index;    
48         return 1;
49     }
50 }
51 
52 //元素出栈,如果栈已经为空返回０，否则返回１
53 int pop(MyStack stack)
54 {
55     if(isEmptyStack(stack))
56         return 0;
57     else
58     {
59         stack ->index--;
60         return 1;
61     }
62 }
63 
64 //取栈顶元素
65 DataType top(MyStack stack)
66 {
67     DataType get;
68     if(!isEmptyStack(stack))
69         get = stack ->ele[stack ->index];
70     return get;
71 }
二叉树特性及详细例子

 

bintree.c

二叉树特性及详细例子
  1 #include "stack.h"
  2 
  3 //访问节点信息
  4 int visit(TreeNode node)
  5 {
  6     printf(" %d", node.element);
  7     return node.nodeIndex;
  8 }
  9 
 10 //先根次序周游
 11 void preOrder(Tree tree, MyStack stack)
 12 {
 13     int index = 0;
 14     TreeNode node;
 15     if(!tree->nodelist[index].isUsed)
 16         return;
 17     push(stack,tree->nodelist[index]);
 18     while(!isEmptyStack(stack))
 19     {
 20         node = top(stack);
 21         pop(stack);
 22         if(node.isUsed == USED)
 23         {
 24             index = visit(node);
 25             push(stack,tree->nodelist[rightChild(tree, index)]);
 26             push(stack,tree->nodelist[leftChild(tree, index)]);
 27         }
 28         
 29     }
 30 }
 31 
 32 //中根次序周游
 33 void inOrder(Tree tree, MyStack stack)
 34 {
 35     int index = 0;
 36     TreeNode node;
 37     if(tree->nodelist[index].isUsed == NOTUSED)
 38         return;
 39     do 
 40     {
 41         while(tree->nodelist[index].isUsed == USED)
 42         {
 43             push(stack,tree->nodelist[index]);
 44             index = leftChild(tree, index);
 45         }
 46         node = top(stack);
 47         pop(stack);
 48         index = visit(node);
 49         index = rightChild(tree,index);
 50     } while(tree->nodelist[index].isUsed == USED || !isEmptyStack(stack));
 51         
 52 }
 53 
 54 //后根次序周游
 55 void postOrder(Tree tree, MyStack stack)
 56 {
 57     int index = 0, rightIndex;
 58     TreeNode node;
 59     if(tree->nodelist[index].isUsed == NOTUSED)
 60         return;
 61     while(tree->nodelist[index].isUsed == USED || !isEmptyStack(stack))
 62     {
 63         while(tree->nodelist[index].isUsed == USED)
 64         {
 65             push(stack,tree->nodelist[index]); 
 66             rightIndex = rightChild(tree, index);
 67             index = leftChild(tree ,index);
 68             if(tree->nodelist[index].isUsed == NOTUSED)
 69                 index = rightIndex;
 70         }
 71         node = top(stack);
 72         pop(stack);
 73         index = visit(node);
 74         //如果栈不为空，且从左子树退回,进入到右子树遍历
 75         if(!isEmptyStack(stack) && index == leftChild(tree,top(stack).nodeIndex))
 76             index = rightChild(tree,top(stack).nodeIndex);
 77         else tree->nodelist[index].isUsed = NOTUSED;
 78     }
 79     
 80 }
 81 
 82 int main()
 83 {
 84     int m = 0, isIn = 1, j, buffer = 0;
 85     Tree tree = NULL;
 86     MyStack stack = NULL;
 87 
 88     printf("请输入节点个数:");
 89     scanf("%d",&m);
 90     tree = createBinTree(m);
 91     stack = createStack(m);
 92     printf("请输入%d个数字:",m);
 93     //将0到5六个元素依次放入二叉树中
 94     for(j = 0; j < m; j++)
 95     {
 96         scanf(" %d", &buffer);
 97         inTree(tree, buffer, isIn);
 98     }
 99 
100     printf("先根次序周游结果为:");
101     preOrder(tree,stack);
102     printf("\n");
103 
104     printf("中根次序周游结果为:");
105     inOrder(tree, stack);
106     printf("\n");
107 
108     printf("后根次序周游结果为:");
109     postOrder(tree, stack);
110     printf("\n");
111 
112     return 1;
113 }
二叉树特性及详细例子

 

按图示二叉树输出:

二叉树特性及详细例子

输出结果为：

二叉树特性及详细例子

 

二叉树连接表示:

二叉树特性及详细例子
  1 #include<stdio.h>
  2 #include<stdlib.h>
  3 typedef int DataType;
  4 typedef struct treenode *TreeNode;
  5 typedef struct treenode *BinTree;
  6 
  7 struct treenode
  8 {
  9     DataType element;
 10     TreeNode llink;
 11     TreeNode rlink;
 12 };
 13 
 14 BinTree createBinTree(DataType x)
 15 {
 16     BinTree tree = NULL;
 17     tree = malloc(sizeof(struct treenode));
 18     tree->element = x;
 19     return tree;
 20 }
 21 
 22 BinTree addToLeft(BinTree t, DataType x)
 23 {
 24     TreeNode node = NULL;
 25     node = malloc(sizeof(struct treenode));
 26     if (node != NULL)
 27         node->element = x;
 28     t -> llink = node;
 29     return node;
 30 }
 31 
 32 BinTree addToRight(BinTree t, DataType x)
 33 {
 34     TreeNode node = NULL;
 35     node = malloc(sizeof(struct treenode));
 36     if (node != NULL)
 37         node->element = x;
 38     t-> rlink = node;
 39     return node;
 40 }
 41 
 42 void visitRoot(BinTree tree)
 43 {
 44     printf("%d ", tree->element);
 45 }
 46 
 47 BinTree leftChild(BinTree tree)
 48 {
 49     return tree->llink;
 50 }
 51 
 52 BinTree rightChild(BinTree tree)
 53 {
 54     return tree->rlink;
 55 }
 56 //用递归方法
 57 //先根周游
 58 void preOrder(BinTree tree)
 59 {
 60     if(tree == NULL)
 61         return;
 62     visitRoot(tree);
 63     preOrder(leftChild(tree));
 64     preOrder(rightChild(tree));
 65 }
 66 
 67 //中根周游
 68 void inOrder(BinTree tree)
 69 {
 70     if(NULL == tree)
 71         return;
 72     inOrder(leftChild(tree));
 73     visitRoot(tree);
 74     inOrder(rightChild(tree));
 75 }
 76 
 77 //后根周游
 78 void postOrder(BinTree tree)
 79 {
 80     if(NULL == tree)
 81         return;
 82     postOrder(leftChild(tree));
 83     postOrder(rightChild(tree));
 84     visitRoot(tree);
 85 }
 86 
 87 int main()
 88 {
 89     BinTree left, right;
 90     BinTree tree = createBinTree(5);
 91     left = addToLeft(tree, 4);
 92     right = addToRight(tree, 3);
 93     addToLeft(left, 8);
 94     addToRight(left, 7);
 95     addToLeft(right, 6);
 96     
 97     printf("先根周游次序：");
 98     preOrder(tree);
 99     printf("\n");
100     printf("中根周游次序：");
101     inOrder(tree);
102     printf("\n");
103     printf("后根周游算法：");
104     postOrder(tree);
105     printf("\n");
106     return 1;
107 }

转载  http://www.cnblogs.com/hanyuan/archive/2012/05/10/bintree.html
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