检查二叉树的所有节点是否可以表示为两个素数之和

给定具有奇数值的N 个节点的二叉树。任务是检查树的所有节点是否可以表示为两个素数之和。

例子：

Input: 

Output: Yes 
Explanation: 
All the nodes in the tree can be represented as the sum of two prime numbers as: 
9 = 2 + 7 
15 = 2 +13 
7 = 2 + 5 
19 = 2 + 17 
25 = 2 + 23 
13 = 11 + 2 
5 = 2 + 3

Input:

Output: No 
Explanation: 
The node with value 27 cannot be represented as the sum of two prime numbers.

方法：

1. 这个想法是使用哥德巴赫的弱猜想，该猜想指出每个大于 5 的奇数都可以表示为三个素数的总和。
2. 要将奇数(例如N )表示为两个素数之和，请将一个素数固定为2 ，如果(N – 2)也是素数，则N可以表示为两个素数之和。
3. 检查树中所有节点的上述条件。如果任何节点不符合上述条件，则打印“否”，否则打印“是”。

C++

// C++ program for the above approach
#include 
using namespace std;
 
// Function to create array to mark
// whether element are prime or not
void spf_array(int arr[], int N)
{
    int i = 0;
 
    // Initially we set same value in
    // array as a index of array.
    for (i = 1; i <= N; i++) {
        arr[i] = i;
    }
 
    // Mark all even elements as 2
    for (i = 2; i <= N; i = i + 2) {
        arr[i] = 2;
    }
 
    // Mark all the multiple of prime
    // numbers as a non-prime
    for (i = 3; i * i <= N; i++) {
        if (arr[i] == i) {
 
            int j = 0;
 
            for (j = i * i; j <= N;
                 j = j + i) {
 
                if (arr[j] == j) {
                    arr[j] = i;
                }
            }
        }
    }
}
 
// Tree Node
struct node {
    int val;
    node* left;
    node* right;
};
 
// Funtion to create node of tree
node* newnode(int i)
{
    node* temp = NULL;
    temp = new node();
    temp->val = i;
    temp->left = NULL;
    temp->right = NULL;
    return temp;
}
 
// Function to check whether the
// tree is prime or not
int prime_tree(node* root, int arr[])
{
    int a = -1;
    if (root != NULL) {
 
        // If element is not the sum of
        // two prime then return 0
        if (root->val <= 3
            || arr[root->val - 2]
                   != root->val - 2) {
 
            return 0;
        }
    }
 
    if (root->left != NULL) {
        a = prime_tree(root->left, arr);
 
        // If a is 0 then we don't need
        // to check further
        if (a == 0) {
            return 0;
        }
    }
 
    if (root->right != NULL) {
 
        a = prime_tree(root->right, arr);
 
        // If a is 0 then we don't need
        // to check further
        if (a == 0) {
            return 0;
        }
    }
 
    return 1;
}
 
// Driver Code
int main()
{
 
    // Given Tree
    node* root = newnode(9);
    root->right = newnode(7);
    root->right->right = newnode(5);
    root->right->left = newnode(13);
    root->left = newnode(15);
    root->left->left = newnode(19);
    root->left->right = newnode(25);
 
    // Number of nodes in the tree
    int n = 50;
 
    // Declare spf[] to store
    // prime numbers
    int brr[n + 1];
    int i = 0;
 
    // Find prime numbers in spf[]
    spf_array(brr, n + 1);
 
    // Function Call
    if (prime_tree(root, brr)) {
        cout << "Yes" << endl;
    }
    else {
        cout << "No" << endl;
    }
}

Java

// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Function to create array to mark
// whether element are prime or not
static void spf_array(int arr[], int N)
{
    int i = 0;
 
    // Initially we set same value in
    // array as a index of array.
    for(i = 1; i <= N; i++)
    {
        arr[i] = i;
    }
 
    // Mark all even elements as 2
    for(i = 2; i <= N; i = i + 2)
    {
        arr[i] = 2;
    }
 
    // Mark all the multiple of prime
    // numbers as a non-prime
    for(i = 3; i * i <= N; i++)
    {
        if (arr[i] == i)
        {
            int j = 0;
            for(j = i * i; j <= N;
                j = j + i)
            {
                if (arr[j] == j)
                {
                    arr[j] = i;
                }
            }
        }
    }
}
 
// Tree Node
static class node
{
    int val;
    node left;
    node right;
};
 
// Funtion to create node of tree
static node newnode(int i)
{
    node temp = null;
    temp = new node();
    temp.val = i;
    temp.left = null;
    temp.right = null;
    return temp;
}
 
// Function to check whether
// the tree is prime or not
static int prime_tree(node root, int arr[])
{
    int a = -1;
    if (root != null)
    {
         
        // If element is not the sum
        // of two prime then return 0
        if (root.val <= 3 ||
         arr[root.val - 2] !=
             root.val - 2)
        {
            return 0;
        }
    }
     
    if (root.left != null)
    {
        a = prime_tree(root.left, arr);
 
        // If a is 0 then we don't
        // need to check further
        if (a == 0)
        {
            return 0;
        }
    }
 
    if (root.right != null)
    {
        a = prime_tree(root.right, arr);
 
        // If a is 0 then we don't
        // need to check further
        if (a == 0)
        {
            return 0;
        }
    }
    return 1;
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given Tree
    node root = newnode(9);
    root.right = newnode(7);
    root.right.right = newnode(5);
    root.right.left = newnode(13);
    root.left = newnode(15);
    root.left.left = newnode(19);
    root.left.right = newnode(25);
 
    // Number of nodes in the tree
    int n = 50;
 
    // Declare spf[] to store
    // prime numbers
    int []brr = new int[n + 1];
    int i = 0;
 
    // Find prime numbers in spf[]
    spf_array(brr, n);
 
    // Function Call
    if (prime_tree(root, brr) == 1)
    {
        System.out.print("Yes" + "\n");
    }
    else
    {
        System.out.print("No" + "\n");
    }
}
}
 
// This code is contributed by Rohit_ranjan

Python3

# Python3 program for the above approach
class Node:
     
    def __init__(self, key):
         
        self.val = key
        self.left = None
        self.right = None
 
# Function to create array to mark
# whether element are prime or not
def spf_array(arr, N):
     
    # Initially we set same value in
    # array as a index of array.
    for i in range(1, N + 1):
        arr[i] = i
 
    # Mark all even elements as 2
    for i in range(2, N + 1, 2):
        arr[i] = 2
 
    # Mark all the multiple of prime
    # numbers as a non-prime
    for i in range(3, N + 1):
        if i * i > N:
            break
         
        if (arr[i] == i):
            for j in range(2 * i, N, i):
                if arr[j] == j:
                    arr[j] = i
 
    return arr
 
# Function to check whether the
# tree is prime or not
def prime_tree(root, arr):
     
    a = -1
     
    if (root != None):
         
        # If element is not the sum of
        # two prime then return 0
        if (root.val <= 3 or 
        arr[root.val - 2] != root.val - 2):
            return 0
 
    if (root.left != None):
        a = prime_tree(root.left, arr)
 
        # If a is 0 then we don't need
        # to check furthe
        if (a == 0):
            return 0
 
    if (root.right != None):
        a = prime_tree(root.right, arr)
 
        # If a is 0 then we don't need
        # to check further
        if (a == 0):
            return 0
 
    return 1
 
# Driver Code
if __name__ == '__main__':
     
    # Given Tree
    root = Node(9);
    root.right = Node(7);
    root.right.right = Node(5);
    root.right.left = Node(13);
    root.left = Node(15);
    root.left.left = Node(19);
    root.left.right = Node(25);
 
    # Number of nodes in the tree
    n = 50
 
    # Declare spf[] to store
    # prime numbers
    arr = [0] * (n + 2)
 
    # Find prime numbers in spf[]
    brr = spf_array(arr, n + 1);
 
    # Function Call
    if (prime_tree(root, brr)):
        print("Yes")
    else:
        print("No")
 
# This code is contributed by mohit kumar 29

C#

// C# program for the above approach
using System;
 
class GFG{
 
// Function to create array to mark
// whether element are prime or not
static void spf_array(int []arr, int N)
{
    int i = 0;
 
    // Initially we set same value in
    // array as a index of array.
    for(i = 1; i <= N; i++)
    {
        arr[i] = i;
    }
 
    // Mark all even elements as 2
    for(i = 2; i <= N; i = i + 2)
    {
        arr[i] = 2;
    }
 
    // Mark all the multiple of prime
    // numbers as a non-prime
    for(i = 3; i * i <= N; i++)
    {
        if (arr[i] == i)
        {
            int j = 0;
            for(j = i * i; j <= N;
                j = j + i)
            {
                if (arr[j] == j)
                {
                    arr[j] = i;
                }
            }
        }
    }
}
 
// Tree Node
class node
{
    public int val;
    public node left;
    public node right;
};
 
// Funtion to create node of tree
static node newnode(int i)
{
    node temp = null;
    temp = new node();
    temp.val = i;
    temp.left = null;
    temp.right = null;
    return temp;
}
 
// Function to check whether
// the tree is prime or not
static int prime_tree(node root, int []arr)
{
    int a = -1;
    if (root != null)
    {
         
        // If element is not the sum
        // of two prime then return 0
        if (root.val <= 3 ||
        arr[root.val - 2] !=
            root.val - 2)
        {
            return 0;
        }
    }
     
    if (root.left != null)
    {
        a = prime_tree(root.left, arr);
 
        // If a is 0 then we don't
        // need to check further
        if (a == 0)
        {
            return 0;
        }
    }
 
    if (root.right != null)
    {
        a = prime_tree(root.right, arr);
 
        // If a is 0 then we don't
        // need to check further
        if (a == 0)
        {
            return 0;
        }
    }
    return 1;
}
 
// Driver Code
public static void Main(String[] args)
{
     
    // Given Tree
    node root = newnode(9);
    root.right = newnode(7);
    root.right.right = newnode(5);
    root.right.left = newnode(13);
    root.left = newnode(15);
    root.left.left = newnode(19);
    root.left.right = newnode(25);
 
    // Number of nodes in the tree
    int n = 50;
 
    // Declare spf[] to store
    // prime numbers
    int []brr = new int[n + 1];
 
    // Find prime numbers in spf[]
    spf_array(brr, n);
 
    // Function Call
    if (prime_tree(root, brr) == 1)
    {
        Console.Write("Yes" + "\n");
    }
    else
    {
        Console.Write("No" + "\n");
    }
}
}
 
// This code is contributed by amal kumar choubey

Javascript

    // javascript program for the above approach
 
    // Function to create array to mark
    // whether element are prime or not
    function spf_array(arr , N) {
        var i = 0;
 
        // Initially we set same value in
        // array as a index of array.
        for (i = 1; i <= N; i++) {
            arr[i] = i;
        }
 
        // Mark all even elements as 2
        for (i = 2; i <= N; i = i + 2) {
            arr[i] = 2;
        }
 
        // Mark all the multiple of prime
        // numbers as a non-prime
        for (i = 3; i * i <= N; i++) {
            if (arr[i] == i) {
                var j = 0;
                for (j = i * i; j <= N; j = j + i) {
                    if (arr[j] == j) {
                        arr[j] = i;
                    }
                }
            }
        }
    }
 
    // Tree Node
    class node {
        constructor(val) {
            this.data = val;
            this.left = null;
            this.right = null;
        }
    }
 
    // Funtion to create node of tree
    function newnode(i) {
        var temp = null;
        temp = new node();
        temp.val = i;
        temp.left = null;
        temp.right = null;
        return temp;
    }
 
    // Function to check whether
    // the tree is prime or not
    function prime_tree( root , arr) {
        var a = -1;
        if (root != null) {
 
            // If element is not the sum
            // of two prime then return 0
            if (root.val <= 3 || arr[root.val - 2] != root.val - 2) {
                return 0;
            }
        }
 
        if (root.left != null) {
            a = prime_tree(root.left, arr);
 
            // If a is 0 then we don't
            // need to check further
            if (a == 0) {
                return 0;
            }
        }
 
        if (root.right != null) {
            a = prime_tree(root.right, arr);
 
            // If a is 0 then we don't
            // need to check further
            if (a == 0) {
                return 0;
            }
        }
        return 1;
    }
 
    // Driver Code
     
 
    // Given Tree
    root = newnode(9);
    root.right = newnode(7);
    root.right.right = newnode(5);
    root.right.left = newnode(13);
    root.left = newnode(15);
    root.left.left = newnode(19);
    root.left.right = newnode(25);
 
    // Number of nodes in the tree
    var n = 50;
 
    // Declare spf to store
    // prime numbers
    var brr = Array(n + 1).fill(0);
    var i = 0;
 
    // Find prime numbers in spf
    spf_array(brr, n);
 
    // Function Call
    if (prime_tree(root, brr) == 1) {
        document.write("Yes" + "\n");
    }
      else {
        document.write("No" + "\n");
    }
 
// This code contributed by gauravrajput1

输出：

Yes

时间复杂度： O (N * log(log N))
辅助空间： O(N)

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