数据结构初阶：二叉树的链式存储结构

该结构对应的树形如下：

三、链式二叉树的基本操作

3.1、前序遍历

访问顺序：根 -> 左 -> 右。

void PreOrder(BTNode* root) {
    if (root == NULL) {
        printf("NULL ");
        return;
    }
    printf("%c ", root->data);
    PreOrder(root->leftChild);
    PreOrder(root->rightChild);
}

递归终止条件是 root 为 NULL。执行时先打印当前节点，再递归处理左右子树。

3.2、中序遍历

访问顺序：左 -> 根 -> 右。

void InOrder(BTNode* root) {
    if (root == NULL) {
        printf("NULL ");
        return;
    }
    InOrder(root->leftChild);
    printf("%c ", root->data);
    InOrder(root->rightChild);
}

3.3、后序遍历

访问顺序：左 -> 右 -> 根。

void LastOrder(BTNode* root) {
    if (root == NULL) {
        printf("NULL ");
        return;
    }
    LastOrder(root->leftChild);
    LastOrder(root->rightChild);
    printf("%c ", root->data);
}

3.4、计算二叉树的总结点个数

采用递归思路：总结点数 = 左子树节点数 + 右子树节点数 + 1。

int BinaryTreeSize_of_Node(BTNode* root) {
    if (root == NULL) {
        return 0;
    }
    return 1 + BinaryTreeSize_of_Node(root->leftChild) + BinaryTreeSize_of_Node(root->rightChild);
}

3.5、计算二叉树的叶子结点个数

叶子节点定义为左右孩子均为空的节点。

int BinaryTreesize_of_Leaf(BTNode* root) {
    if (root == NULL) return 0;
    if (root->leftChild == NULL && root->rightChild == NULL) return 1;
    return BinaryTreesize_of_Leaf(root->leftChild) + BinaryTreesize_of_Leaf(root->rightChild);
}

3.6、计算第 k 层节点个数

递归统计左、右子树第 k-1 层的节点数之和。

int BinaryTreeSize_of_K(BTNode* root, int k) {
    if (root == NULL) return 0;
    if (k == 1) return 1;
    return BinaryTreeSize_of_K(root->leftChild, k - 1) + BinaryTreeSize_of_K(root->rightChild, k - 1);
}

3.7、计算二叉树的深度

深度取决于左右子树深度的最大值加 1。

int BinaryTreeSize_of_Depth(BTNode* root) {
    if (root == NULL) {
        return 0;
    }
    int leftDepth = BinaryTreeSize_of_Depth(root->leftChild);
    int rightDepth = BinaryTreeSize_of_Depth(root->rightChild);
    return 1 + (leftDepth > rightDepth ? leftDepth : rightDepth);
}

3.8、查找元素

遍历整棵树查找指定值。

void BinaryTreeSearch_of_val(BTNode* root, Elemtype val) {
    if (root == NULL) return;
    if (root->data == val) {
        printf("找到了!\n");
        return;
    }
    BinaryTreeSearch_of_val(root->leftChild, val);
    BinaryTreeSearch_of_val(root->rightChild, val);
}

3.9、层序遍历

借助队列实现，按层级从左到右访问。

void LevelOrder(BTNode* root) {
    Queue q;
    Queue* pQueue = &q;
    InitQueue(pQueue);
    Push_Queue(pQueue, root);
    while (!isEmpty(pQueue)) {
        BTNode* top = GetElemFront(pQueue);
        printf("%c ", top->data);
        Pop_Queue(pQueue);
        if (top->leftChild) {
            Push_Queue(pQueue, top->leftChild);
        }
        if (top->rightChild) {
            Push_Queue(pQueue, top->rightChild);
        }
    }
    DestroyQueue(pQueue);
}

3.10、判断是否为完全二叉树

利用层序遍历，遇到空节点后若还有非空节点，则不是完全二叉树。

bool IsBinaryTree_of_Complete(BTNode* root) {
    Queue q;
    Queue* pQueue = &q;
    InitQueue(pQueue);
    Push_Queue(pQueue, root);
    while (!isEmpty(pQueue)) {
        QElemtype top = GetElemFront(pQueue);
        Pop_Queue(pQueue);
        if (top == NULL) {
            break;
        }
        Push_Queue(pQueue, top->leftChild);
        Push_Queue(pQueue, top->rightChild);
    }
    while (!isEmpty(pQueue)) {
        BTNode* top = GetElemFront(pQueue);
        Pop_Queue(pQueue);
        if (top != NULL) {
            DestroyQueue(pQueue);
            return false;
        }
    }
    DestroyQueue(pQueue);
    return true;
}

四、综合代码

整合上述功能，包含主函数测试及队列辅助代码。

二叉树核心代码

#define _CRT_SECURE_NO_WARNINGS 1
#include<stdio.h>
#include<stdlib.h>
#include<assert.h>
#include"Queue.h"

typedef char Elemtype;

typedef struct BinaryTree {
    Elemtype data;
    struct BinaryTree* leftChild;
    struct BinaryTree* rightChild;
} BTNode;

BTNode* BuyNode(Elemtype data) {
    BTNode* newNode = (BTNode*)malloc(sizeof(BTNode));
    if (newNode == NULL) {
        perror("create new node fail!\n");
    }
    newNode->data = data;
    newNode->leftChild = newNode->rightChild = NULL;
    return newNode;
}

BTNode* createNewBT() {
    BTNode* NodeA = BuyNode('A');
    BTNode* NodeB = BuyNode('B');
    BTNode* NodeC = BuyNode('C');
    BTNode* NodeD = BuyNode('D');
    BTNode* NodeE = BuyNode('E');
    BTNode* NodeF = BuyNode('F');
    BTNode* NodeG = BuyNode('G');
    BTNode* NodeH = BuyNode('H');
    BTNode* NodeI = BuyNode('I');
    BTNode* NodeO = BuyNode('O');
    NodeA->leftChild = NodeB;   NodeA->rightChild = NodeC;
    NodeB->leftChild = NodeD;   NodeB->rightChild = NodeE;
    NodeD->leftChild = NodeH;   NodeC->leftChild = NodeF;
    NodeF->leftChild = NodeI;   NodeH->leftChild = NodeO;
    return NodeA;
}

void PreOrder(BTNode* root) {
    if (root == NULL) { printf("NULL "); return; }
    printf("%c ", root->data); PreOrder(root->leftChild); PreOrder(root->rightChild);
}

void InOrder(BTNode* root) {
    if (root == NULL) { printf("NULL "); return; }
    InOrder(root->leftChild); printf("%c ", root->data); InOrder(root->rightChild);
}

void LastOrder(BTNode* root) {
    if (root == NULL) { printf("NULL "); return; }
    LastOrder(root->leftChild); LastOrder(root->rightChild); printf("%c ", root->data);
}

int BinaryTreeSize_of_Node(BTNode* root) {
    if (root == NULL) return 0;
    return 1 + BinaryTreeSize_of_Node(root->leftChild) + BinaryTreeSize_of_Node(root->rightChild);
}

int BinaryTreesize_of_Leaf(BTNode* root) {
    if (root == NULL) return 0;
    if (root->leftChild == NULL && root->rightChild == NULL) return 1;
    return BinaryTreesize_of_Leaf(root->leftChild) + BinaryTreesize_of_Leaf(root->rightChild);
}

int BinaryTreeSize_of_K(BTNode* root, int k) {
    if (root == NULL) return 0;
    if (k == 1) return 1;
    return BinaryTreeSize_of_K(root->leftChild, k - 1) + BinaryTreeSize_of_K(root->rightChild, k - 1);
}

int BinaryTreeSize_of_Depth(BTNode* root) {
    if (root == NULL) return 0;
    int leftDepth = BinaryTreeSize_of_Depth(root->leftChild);
    int rightDepth = BinaryTreeSize_of_Depth(root->rightChild);
    return 1 + (leftDepth > rightDepth ? leftDepth : rightDepth);
}

void BinaryTreeSearch_of_val(BTNode* root, Elemtype val) {
    if (root == NULL) return;
    if (root->data == val) { printf("找到了!\n"); return; }
    BinaryTreeSearch_of_val(root->leftChild, val);
    BinaryTreeSearch_of_val(root->rightChild, val);
}

void LevelOrder(BTNode* root) {
    Queue q; Queue* pQueue = &q; InitQueue(pQueue);
    Push_Queue(pQueue, root);
    while (!isEmpty(pQueue)) {
        BTNode* top = GetElemFront(pQueue);
        printf("%c ", top->data); Pop_Queue(pQueue);
        if (top->leftChild) Push_Queue(pQueue, top->leftChild);
        if (top->rightChild) Push_Queue(pQueue, top->rightChild);
    }
    DestroyQueue(pQueue);
}

bool IsBinaryTree_of_Complete(BTNode* root) {
    Queue q; Queue* pQueue = &q; InitQueue(pQueue);
    Push_Queue(pQueue, root);
    while (!isEmpty(pQueue)) {
        QElemtype top = GetElemFront(pQueue); Pop_Queue(pQueue);
        if (top == NULL) break;
        Push_Queue(pQueue, top->leftChild); Push_Queue(pQueue, top->rightChild);
    }
    while (!isEmpty(pQueue)) {
        BTNode* top = GetElemFront(pQueue); Pop_Queue(pQueue);
        if (top != NULL) { DestroyQueue(pQueue); return false; }
    }
    DestroyQueue(pQueue); return true;
}

int main() {
    BTNode* NodeA = createNewBT();
    printf("前序遍历:"); PreOrder(NodeA); printf("\n");
    printf("中序遍历:"); InOrder(NodeA); printf("\n");
    printf("后序遍历:"); LastOrder(NodeA); printf("\n");
    printf("总结点：%d\n", BinaryTreeSize_of_Node(NodeA));
    printf("叶子节点：%d\n", BinaryTreesize_of_Leaf(NodeA));
    printf("第 4 层节点：%d\n", BinaryTreeSize_of_K(NodeA, 4));
    printf("深度：%d\n", BinaryTreeSize_of_Depth(NodeA));
    printf("层序遍历:"); LevelOrder(NodeA); printf("\n");
    printf("是否完全二叉树：%s\n", IsBinaryTree_of_Complete(NodeA) ? "是" : "否");
    return 0;
}

队列辅助代码

#pragma once
#include<stdio.h>
#include<stdlib.h>
#include<assert.h>
#include<stdbool.h>

typedef struct BinaryTree* QElemtype;
int count = 0;

typedef struct QueueNode {
    QElemtype data;
    struct QueueNode* next;
} QueueNode;

typedef struct Queue {
    QueueNode* phead;
    QueueNode* ptail;
} Queue;

void InitQueue(Queue* pQueue) {
    pQueue->phead = pQueue->ptail = NULL;
}

void Push_Queue(Queue* pQueue, QElemtype x) {
    QueueNode* newNode = (QueueNode*)malloc(sizeof(QueueNode));
    if (newNode == NULL) { printf("创建新节点失败!!!\n"); exit(0); }
    newNode->data = x; newNode->next = NULL;
    if (pQueue->phead == NULL) {
        pQueue->phead = pQueue->ptail = newNode;
    } else {
        pQueue->ptail->next = newNode;
        pQueue->ptail = newNode;
    }
    count++;
}

bool isEmpty(Queue* pQueue) {
    assert(pQueue);
    return pQueue->phead == NULL;
}

void Pop_Queue(Queue* pQueue) {
    assert(pQueue);
    if (pQueue->phead == NULL) { printf("队列为空!!!\n"); exit(0); }
    QueueNode* next = pQueue->phead->next;
    free(pQueue->phead);
    pQueue->phead = next;
    count--;
}

QElemtype GetElemFront(Queue* pQueue) {
    assert(pQueue);
    if (isEmpty(pQueue)) exit(1);
    return pQueue->phead->data;
}

void DestroyQueue(Queue* pQueue) {
    assert(pQueue);
    QueueNode* pcur = pQueue->phead;
    while (pcur != NULL) {
        QueueNode* next = pcur->next;
        free(pcur);
        pcur = next;
    }
    pQueue->phead = pQueue->ptail = NULL;
}

运行结果将展示各遍历序列及统计信息。