数据结构 二叉树

二叉树性质

n 性质1: 在二叉树的第i层上至多有2^i-1个结点（i>0）

n 性质2: 深度为k的二叉树至多有2^k-1个结点（k>0）

n 性质3: 对于任何一棵二叉树，若度为2的结点数有n2个，则叶子数（n0）必定为n2＋1 （即n0=n2+1）

 

二叉树的递归遍历源码：分为先序遍历：根 左 右 ；中序遍历：左 根 右；后续遍历：左 右 根

#define _CRT_SECURE_NO_WARNINGS
#include<stdio.h>
#include<string.h>
#include<stdlib.h>

//二叉树节点
struct BiNode{
    char ch;
    struct BiNode *lchild;
    struct BiNode *rchild;
};

//二叉树的递归遍历先序
void recursion(struct BiNode *root){

    if (NULL == root){
        return;
    }

    //再访问左子树
    recursion(root->lchild);//printf 先序遍历
    //再访问右子树
    recursion(root->rchild);// printf 中顺序遍历
    //先访问根节点
    printf("%c ", root->ch);// printf  后续遍历
}

void test(){

    //二叉树节点
    struct BiNode node1 = { 'A', NULL, NULL };
    struct BiNode node2 = { 'B', NULL, NULL };
    struct BiNode node3 = { 'C', NULL, NULL };
    struct BiNode node4 = { 'D', NULL, NULL };
    struct BiNode node5 = { 'E', NULL, NULL };
    struct BiNode node6 = { 'F', NULL, NULL };
    struct BiNode node7 = { 'G', NULL, NULL };
    struct BiNode node8 = { 'H', NULL, NULL };

    node1.lchild = &node2;
    node1.rchild = &node6;
    node2.rchild = &node3;
    node3.lchild = &node4;
    node3.rchild = &node5;
    node6.rchild = &node7;
    node7.lchild = &node8;

    //遍历二叉树
    recursion(&node1);
}

int main(){

    test();

    system("pause");
    return EXIT_SUCCESS;
}

通过栈实现二叉树的非递归遍历：

见数据结构 栈的实现

#define _CRT_SECURE_NO_WARNINGS
#include<stdio.h>
#include<string.h>
#include<stdlib.h>
#include"SeqStack.h"
#include<stdbool.h>

//二叉树节点
struct BiNode{
    char ch;
    struct BiNode *lchild;
    struct BiNode *rchild;
};

struct Info{
    struct BiNode *node;
    bool flag;
};

struct Info *CreateInfo(struct BiNode *node,bool flag){
    struct Info *info = malloc(sizeof(struct Info));
    info->node = node;
    info->flag = flag;

    return info;
}


void nonrecursion(struct BiNode *root){
    
    //初始化stack
    void *stack = Init_SeqStack();
    //1. 先把根节点入stack
    Push_SeqStack(stack, CreateInfo(root, false));

    while (Size_SeqStack(stack) > 0){
        
        //2. 获得stack顶元素
        struct Info *info = (struct Info *)Top_SeqStack(stack);
        Pop_SeqStack(stack);

        //3. 判断当前节点的flag是否为true,true:直接输出。false:左子树和右子树分别压stack，当前节点二次入stack
        if(info->flag){
            printf("%c ",info->node->ch);
            free(info);
            continue;
        }


        
        //右子树入stack
        if (info->node->rchild != NULL){
            Push_SeqStack(stack, CreateInfo(info->node->rchild, false));
        }

        //根节点入stack
        info->flag = true;
        Push_SeqStack(stack, info);


        //左子树入stack
        if (info->node->lchild != NULL){
            Push_SeqStack(stack, CreateInfo(info->node->lchild, false));
        }


    }

}

void test(){

    //二叉树节点
    struct BiNode node1 = { 'A', NULL, NULL };
    struct BiNode node2 = { 'B', NULL, NULL };
    struct BiNode node3 = { 'C', NULL, NULL };
    struct BiNode node4 = { 'D', NULL, NULL };
    struct BiNode node5 = { 'E', NULL, NULL };
    struct BiNode node6 = { 'F', NULL, NULL };
    struct BiNode node7 = { 'G', NULL, NULL };
    struct BiNode node8 = { 'H', NULL, NULL };

    node1.lchild = &node2;
    node1.rchild = &node6;
    node2.rchild = &node3;
    node3.lchild = &node4;
    node3.rchild = &node5;
    node6.rchild = &node7;
    node7.lchild = &node8;

    nonrecursion(&node1);

}



int main(){

    test();


    

    system("pause");
    return EXIT_SUCCESS;
}

求二叉数的叶子节点函数：

void caculateLeafNums(struct BiNode *root,int *num){

    if (NULL == root){
        return;
    }

    if (root->lchild == NULL && root->rchild == NULL){
        (*num)++;
    }
    
    //遍历左子树
    caculateLeafNums(root->lchild, num);
    //遍历右子树
    caculateLeafNums(root->rchild, num);

}

二叉树的拷贝、释放、二叉树高度函数：

//拷贝二叉树
struct BiNode *copyTree(struct BiNode *root){

    if (NULL == root){
        return NULL;
    }

    //递归拷贝左子树
    struct BiNode *lchild = copyTree(root->lchild);
    //递归拷贝右子树
    struct BiNode *rchild = copyTree(root->rchild);

    //拷贝当前节点
    struct BiNode *newnode = malloc(sizeof(struct BiNode));
    printf("%c ",root->ch);
    newnode->ch = root->ch;
    newnode->lchild = lchild;
    newnode->rchild = rchild;

    return newnode;
}

//释放二叉树内存
void freeSpace(struct BiNode *root){
    if (NULL == root){
        return;
    }

    //释放左子树
    freeSpace(root->lchild);
    //释放右子树
    freeSpace(root->rchild);
    //释放根节点
    free(root);

}

//递归遍历
void foreach(struct BiNode *root){
    if (NULL == root){
        return;
    }

    printf("%c ",root->ch);
    foreach(root->lchild);
    foreach(root->rchild);
}

//求树的高度
int caculateTreeHeight(struct BiNode *root){

    if (NULL == root){
        return 0;
    }


    //求左子树的高度
    int lheight = caculateTreeHeight(root->lchild);
    //求右子树的高度
    int rheight = caculateTreeHeight(root->rchild);

    int height = lheight > rheight ? lheight + 1 : rheight + 1;

    return height;
}