在二叉树中找到从根到叶的最大 GCD 值

给定一棵二叉树，任务是从根节点到叶节点的任何路径中找到GCD的最大值。

例子：

Input: Below is the given tree:

Output: 3
Explanation:
Path 1: 15->3->5 = gcd(15, 3, 15) =3
Path 2: 15->3->1 =gcd(15, 3, 1) = 1
Path 3: 15->7->31=gcd(15, 7, 31)= 1
Path 4: 15->7->9 = gcd(15, 7, 9) =1, out of these 3 is the maximum.

Input: Below is the given tree:

Output: 1

方法：思想是遍历从根节点到叶子节点的所有路径，并计算该路径中出现的所有节点的GCD。以下是步骤：

1. 对给定的二叉树执行预序遍历。
2. 迭代所有路径并跟踪数组中的所有路径值。
3. 每当遇到叶值时，就找到数组中所有值的 GCD。
4. 将 GCD 更新为最大值。

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
using namespace std;
 
// Initialise to update the maximum
// gcd value from all the path
int maxm = 0;
 
// Node structure
struct Node {
    int val;
 
    // Left & right child of the node
    Node *left, *right;
 
    // Initialize constructor
    Node(int x)
    {
        val = x;
        left = NULL;
        right = NULL;
    }
};
 
// Function to find gcd of a and b
int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}
 
// function to find the gcd of a path
int find_gcd(vector arr)
{
    if (arr.size() == 1)
        return arr[0];
 
    int g = arr[0];
 
    for (int i = 1; i < arr.size(); i++) {
        g = gcd(g, arr[i]);
    }
 
    return g;
}
 
// Function to find the maximum value
// of gcd from root to leaf
// in a Binary tree
void maxm_gcd(Node* root, vector ans)
{
    // Check if root is not null
    if (!root)
        return;
 
    if (root->left == NULL
        and root->right == NULL) {
        ans.push_back(root->val);
 
        // Find the maxmimum gcd of
        // path value and store in
        // global maxm variable
        maxm = max(find_gcd(ans),
                   maxm);
 
        return;
    }
 
    // Traverse left of binary tree
    ans.push_back(root->val);
    maxm_gcd(root->left, ans);
 
    // Traverse right of the binary tree
    maxm_gcd(root->right, ans);
}
 
// Driver Code
int main()
{
    // Given Tree
    Node* root = new Node(15);
    root->left = new Node(3);
    root->right = new Node(7);
    root->left->left = new Node(15);
    root->left->right = new Node(1);
    root->right->left = new Node(31);
    root->right->right = new Node(9);
 
    // Function Call
    maxm_gcd(root, {});
 
    // Print the maximum AND value
    cout << maxm << endl;
 
    return 0;
}

Java

// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Initialise to update the maximum
// gcd value from all the path
static int maxm = 0;
 
// Node structure
static class Node
{
    int val;
 
    // Left & right child of the node
    Node left, right;
 
    // Initialize constructor
    Node(int x)
    {
        val = x;
        left = null;
        right = null;
    }
};
 
// Function to find gcd of a and b
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}
 
// Function to find the gcd of a path
static int find_gcd(Vector arr)
{
    if (arr.size() == 1)
        return arr.get(0);
 
    int g = arr.get(0);
 
    for(int i = 1; i < arr.size(); i++)
    {
        g = gcd(g, arr.get(1));
    }
    return g;
}
 
// Function to find the maximum value
// of gcd from root to leaf
// in a Binary tree
static void maxm_gcd(Node root,
                     Vector ans)
{
     
    // Check if root is not null
    if (root == null)
        return;
 
    if (root.left == null &&
        root.right == null)
    {
        ans.add(root.val);
 
        // Find the maxmimum gcd of
        // path value and store in
        // global maxm variable
        maxm = Math.max(find_gcd(ans),
                        maxm);
                         
        return;
    }
 
    // Traverse left of binary tree
    ans.add(root.val);
    maxm_gcd(root.left, ans);
 
    // Traverse right of the binary tree
    maxm_gcd(root.right, ans);
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given Tree
    Node root = new Node(15);
    root.left = new Node(3);
    root.right = new Node(7);
    root.left.left = new Node(15);
    root.left.right = new Node(1);
    root.right.left = new Node(31);
    root.right.right = new Node(9);
 
    // Function call
    maxm_gcd(root, new Vector<>());
 
    // Print the maximum AND value
    System.out.print(maxm + "\n");
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 implementation of
# the above approach
 
# Initialise to update the maximum
# gcd value from all the path
global maxm
maxm = 0
 
# Node structure
class Node:
 
    # Initialize constructor
    def __init__(self, x):
 
        self.val = x
        self.left = None
        self.right = None
 
# Function to find gcd of a and b
def gcd(a, b):
 
    if(b == 0):
        return a
    return gcd(b, a % b)
 
# Function to find the gcd of a path
def find_gcd(arr):
 
    if(len(arr) == 1):
        return arr[0]
 
    g = arr[0]
 
    for i in range(1, len(arr)):
        g = gcd(g, arr[i])
 
    return g
 
# Function to find the maximum value
# of gcd from root to leaf
# in a Binary tree
def maxm_gcd(root, ans):
 
    global maxm
 
    # Check if root is not null
    if(not root):
        return
 
    if(root.left == None and
       root.right == None):
        ans.append(root.val)
 
        # Find the maxmimum gcd of
        # path value and store in
        # global maxm variable
        maxm = max(find_gcd(ans), maxm)
 
        return
 
    # Traverse left of binary tree
    ans.append(root.val)
    maxm_gcd(root.left, ans)
 
    # Traverse right of the binary tree
    maxm_gcd(root.right, ans)
 
# Driver Code
if __name__ == '__main__':
 
    # Given Tree
    root = Node(15)
    root.left = Node(3)
    root.right = Node(7)
    root.left.left = Node(15)
    root.left.right = Node(1)
    root.right.left = Node(31)
    root.right.right = Node(9)
 
    # Function call
    maxm_gcd(root, [])
     
    # Print the maximum AND value
    print(maxm)
 
# This code is contributed by Shivam Singh

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG{
 
// Initialise to update the maximum
// gcd value from all the path
static int maxm = 0;
 
// Node structure
class Node
{
    public int val;
 
    // Left & right child of the node
    public Node left, right;
 
    // Initialize constructor
    public Node(int x)
    {
        val = x;
        left = null;
        right = null;
    }
};
 
// Function to find gcd of a and b
static int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}
 
// Function to find the gcd of a path
static int find_gcd(List arr)
{
    if (arr.Count == 1)
        return arr[0];
 
    int g = arr[0];
 
    for(int i = 1; i < arr.Count; i++)
    {
        g = gcd(g, arr[1]);
    }
    return g;
}
 
// Function to find the maximum value
// of gcd from root to leaf
// in a Binary tree
static void maxm_gcd(Node root,
                     List ans)
{
     
    // Check if root is not null
    if (root == null)
        return;
 
    if (root.left == null &&
        root.right == null)
    {
        ans.Add(root.val);
 
        // Find the maxmimum gcd of
        // path value and store in
        // global maxm variable
        maxm = Math.Max(find_gcd(ans),
                                 maxm);
                         
        return;
    }
 
    // Traverse left of binary tree
    ans.Add(root.val);
    maxm_gcd(root.left, ans);
 
    // Traverse right of the binary tree
    maxm_gcd(root.right, ans);
}
 
// Driver Code
public static void Main(String[] args)
{
     
    // Given Tree
    Node root = new Node(15);
    root.left = new Node(3);
    root.right = new Node(7);
    root.left.left = new Node(15);
    root.left.right = new Node(1);
    root.right.left = new Node(31);
    root.right.right = new Node(9);
 
    // Function call
    maxm_gcd(root, new List());
 
    // Print the maximum AND value
    Console.Write(maxm + "\n");
}
}
 
// This code is contributed by sapnasingh4991

Javascript

 
// JavaScript program for the above approach
 
// Initialise to update the maximum
// gcd value from all the path
let maxm = 0;
 
// Node structure
class Node
{
 
    // Initialize constructor
    constructor(x)
    {
        this.val = x;
        this.left = null;
        this.right = null;
    }
}
 
var root;
 
// Function to find gcd of a and b
function gcd(a, b)
{
    if (b == 0)
        return a;
         
    return gcd(b, a % b);
}
 
// Function to find the gcd of a path
function find_gcd(arr)
{
    if (arr.length == 1)
        return arr[0];
 
    let g = arr[0];
 
    for(let i = 1; i < arr.length; i++)
    {
        g = gcd(g, arr[1]);
    }
    return g;
}
 
// Function to find the maximum value
// of gcd from root to leaf
// in a Binary tree
function maxm_gcd(root, ans)
{
     
    // Check if root is not null
    if (root == null)
        return;
 
    if (root.left == null &&
        root.right == null)
    {
        ans.push(root.val);
 
        // Find the maxmimum gcd of
        // path value and store in
        // global maxm variable
        maxm = Math.max(find_gcd(ans),
                        maxm);
                         
        return;
    }
 
    // Traverse left of binary tree
    ans.push(root.val);
    maxm_gcd(root.left, ans);
 
    // Traverse right of the binary tree
    maxm_gcd(root.right, ans);
}
 
// Driver Code
 
// Given Tree
root = new Node(15);
root.left = new Node(3);
root.right = new Node(7);
root.left.left = new Node(15);
root.left.right = new Node(1);
root.right.left = new Node(31);
root.right.right = new Node(9);
 
// Function call
let arr = [];
maxm_gcd(root, arr);
 
// Print the maximum AND value
document.write(maxm);
 
// This code is contributed by Dharanendra L V.
 

输出：

3

时间复杂度： O(N ² )
辅助空间： O(N)

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