判断二叉树是否为满二叉树

1、题目

给定一棵二叉树的根节点，判断这棵二叉树是否为满二叉树。

2、分析

满二叉树：如果树的高度是 $h$，则节点数一定是 $2^h-1$。

所以根据二叉树的递归套路：假设X为根节点的树，向左右子树收集两个信息：高度 $h$ 和 节点数 $n$，判断是否满足满二叉树的条件。

另外，如果左树是满二叉树，右树是满二叉树，且左右树的高度一致，那么整棵树是满二叉树。

所以，也可以利用二叉树的递归套路，从左右树获取信息：（1）是否为满二叉树 （2）高度

3、实现

C++版

/************************************************************************* > File Name: 036.判断二叉树是否为满二叉树.cpp > Author: Maureen > Mail: [email protected] > Created Time: 五 7/ 1 16:37:02 2022 ************************************************************************/

#include <iostream>
#include <ctime>
using namespace std;

class TreeNode {
    
public:
    int value;
    TreeNode *left;
    TreeNode *right;

    TreeNode(int data) : value(data) {
    }
};


//方法1：收集整棵树的高度和节点数 
//只有满二叉树满足：2^h - 1 == n
class Info1 {
    
public:
    int height;//高度
    int nodeCnt;//节点数

    Info1(int h, int c) : height(h), nodeCnt(c) {
    }
};


Info1 *process1(TreeNode *x) {
    
    if (x == nullptr) {
    
        return new Info1(0, 0);
    }

    Info1 *leftInfo = process1(x->left);
    Info1 *rightInfo = process1(x->right);

    int height = max(leftInfo->height, rightInfo->height) + 1;
    int nodeCnt = leftInfo->nodeCnt + rightInfo->nodeCnt + 1;

    return new Info1(height, nodeCnt);
}

bool isFullBT1(TreeNode *root) {
    
    if (root == nullptr) return true;
    
    Info1 *info = process1(root);

    return (1 << info->height) - 1 == info->nodeCnt;
}


//方法2: 收集 子树是否满二叉树 和 子树高度 
//左树满 && 右树满 && 左树高度和右树高度一样，那么整棵树就是满的
class Info2 {
    
public:
    bool isFull;
    int height;

    Info2(bool isfull, int h) : isFull(isfull), height(h) {
    }
};

Info2 *process2(TreeNode *x) {
    
    if (x == nullptr) {
    
        return new Info2(true, 0);
    }

    Info2 *leftInfo = process2(x->left);
    Info2 *rightInfo = process2(x->right);

    int height = max(leftInfo->height, rightInfo->height) + 1;
    bool isFull = leftInfo->isFull && rightInfo->isFull && leftInfo->height == rightInfo->height;

    return new Info2(isFull, height);
}


bool isFullBT2(TreeNode *root) {
    
    return process2(root)->isFull;
}


//for test 
TreeNode *generate(int level, int maxl, int maxv) {
    
    if (level > maxl || (rand() % 100 / (double)101) < 0.5)
        return nullptr;

    TreeNode *root = new TreeNode(rand() % maxv);
    root->left = generate(level + 1, maxl, maxv);
    root->right = generate(level + 1, maxl, maxv);

    return root;
}

TreeNode *generateRandomBST(int level, int value) {
    
    return generate(1, level, value);
}

int main() {
    
    srand(time(0));

    int level = 5;
    int value = 100;
    int testTimes = 1000001;

    cout << "Test begin:" << endl;
    for (int i = 0; i < testTimes; i++) {
    
        TreeNode *root = generateRandomBST(level, value);
        if (isFullBT1(root) != isFullBT2(root)) {
    
            cout << "Failed!" << endl;
            return 0;
        }
    }
    cout << "Success!" << endl;
    return 0;
}

Java 版

public class isFullBT {
    
    public static class Node {
    
		public int value;
		public Node left;
		public Node right;

		public Node(int data) {
    
			this.value = data;
		}
	}    

	// 第一种方法
	// 收集整棵树的高度h，和节点数n
	// 只有满二叉树满足 : 2 ^ h - 1 == n
	public static boolean isFull1(Node head) {
    
		if (head == null) {
    
			return true;
		}
		Info1 all = process1(head);
		return (1 << all.height) - 1 == all.nodes;
	}

	public static class Info1 {
    
		public int height; //高度
		public int nodes; //节点数

		public Info1(int h, int n) {
    
			height = h;
			nodes = n;
		}
	}

	public static Info1 process1(Node head) {
    
		if (head == null) {
    
			return new Info1(0, 0);
		}
		Info1 leftInfo = process1(head.left);
		Info1 rightInfo = process1(head.right);
		int height = Math.max(leftInfo.height, rightInfo.height) + 1;
		int nodes = leftInfo.nodes + rightInfo.nodes + 1;
		return new Info1(height, nodes);
	}

	// 第二种方法
	// 收集子树是否是满二叉树
	// 收集子树的高度
	// 左树满 && 右树满 && 左右树高度一样 -> 整棵树是满的
	public static boolean isFull2(Node head) {
    
		if (head == null) {
    
			return true;
		}
		return process2(head).isFull;
	}

	public static class Info2 {
    
		public boolean isFull; 
		public int height;

		public Info2(boolean f, int h) {
    
			isFull = f;
			height = h;
		}
	}

	public static Info2 process2(Node h) {
    
		if (h == null) {
    
			return new Info2(true, 0);
		}
		Info2 leftInfo = process2(h.left);
		Info2 rightInfo = process2(h.right);
		boolean isFull = leftInfo.isFull && rightInfo.isFull && leftInfo.height == rightInfo.height;
		int height = Math.max(leftInfo.height, rightInfo.height) + 1;
		return new Info2(isFull, height);
	}

	// for test
	public static Node generateRandomBST(int maxLevel, int maxValue) {
    
		return generate(1, maxLevel, maxValue);
	}

	// for test
	public static Node generate(int level, int maxLevel, int maxValue) {
    
		if (level > maxLevel || Math.random() < 0.5) {
    
			return null;
		}
		Node head = new Node((int) (Math.random() * maxValue));
		head.left = generate(level + 1, maxLevel, maxValue);
		head.right = generate(level + 1, maxLevel, maxValue);
		return head;
	}

	public static void main(String[] args) {
    
		int maxLevel = 5;
		int maxValue = 100;
		int testTimes = 1000000;
		System.out.println("测试开始");
		for (int i = 0; i < testTimes; i++) {
    
			Node head = generateRandomBST(maxLevel, maxValue);
			if (isFull1(head) != isFull2(head)) {
    
				System.out.println("出错了!");
			}
		}
		System.out.println("测试结束");
	}    
}