Print the nodes with a prime degree in given Prufer sequence of a Tree

Given a Prufer sequence of a Tree, the task is to print the nodes with prime-degree in this tree.

Examples:

Input: arr[] = {4, 1, 3, 4} 
Output: 1 3 4
Explanation:
The tree is:
2----4----3----1----5
     |
     6 
Hence, the degree of 1, 3 and 4
are 2, 2 and 3 respectively
which are prime.

Input: a[] = {1, 2, 2} 
Output: 1 2

Approach:

  1. Since the length of prufer sequence is N – 2 if N is the number of nodes. Therefore create an array degree[] of size 2 more than the length of the Prufer sequence.
  2. Initially, fill the degree array with 1.
  3. Iterate in the Prufer sequence and increase the frequency in the degree table for every element. This method works because the frequency of a node in the Prufer sequence is one less than the degree in the tree.
  4. Further, to check if the node degree is prime or not, we will use Sieve Of eratosthenes. Create a sieve which will help us to identify if the degree is prime or not in O(1) time.
  5. If a node has a prime degree, then print the node number.

Below is the implementation of the above approach:

C++

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// C++ implementation to print the 
// nodes with prime degree from the
// given prufer sequence
  
#include <bits/stdc++.h>
  
using namespace std;
  
// Function to create Sieve
// to check primes
void SieveOfEratosthenes(
       bool prime[], int p_size)
{
    // False here indicates
    // that it is not prime
    prime[0] = false;
    prime[1] = false;
  
    for (int p = 2; p * p <= p_size; p++) {
  
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p]) {
  
            // Update all multiples of p,
            // set them to non-prime
            for (int i = p * 2; i <= p_size; 
                                      i += p)
                prime[i] = false;
        }
    }
}
  
// Function to print the nodes with
// prime degree in the tree
// whose Prufer sequence is given
void PrimeDegreeNodes(int prufer[], int n)
{
    int nodes = n + 2;
  
    bool prime[nodes + 1];
    memset(prime, true, sizeof(prime));
  
    SieveOfEratosthenes(prime, nodes + 1);
  
    // Hash-table to mark the
    // degree of every node
    int degree[n + 2 + 1];
  
    // Initially let all the degrees be 1
    for (int i = 1; i <= nodes; i++)
        degree[i] = 1;
  
    // Increase the count of the degree
    for (int i = 0; i < n; i++)
        degree[prufer[i]]++;
  
    // Print the nodes with prime degree
    for (int i = 1; i <= nodes; i++) {
        if (prime[degree[i]]) {
            cout << i << " ";
        }
    }
}
  
// Driver Code
int main()
{
    int a[] = { 4, 1, 3, 4 };
    int n = sizeof(a) / sizeof(a[0]);
  
    PrimeDegreeNodes(a, n);
  
    return 0;
}

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Java

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// Java implementation to print the 
// nodes with prime degree from the
// given prufer sequence
   
  
   
import java.util.*;
  
class GFG{
   
// Function to create Sieve
// to check primes
static void SieveOfEratosthenes(
       boolean prime[], int p_size)
{
    // False here indicates
    // that it is not prime
    prime[0] = false;
    prime[1] = false;
   
    for (int p = 2; p * p <= p_size; p++) {
   
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p]) {
   
            // Update all multiples of p,
            // set them to non-prime
            for (int i = p * 2; i <= p_size; 
                                      i += p)
                prime[i] = false;
        }
    }
}
   
// Function to print the nodes with
// prime degree in the tree
// whose Prufer sequence is given
static void PrimeDegreeNodes(int prufer[], int n)
{
    int nodes = n + 2;
   
    boolean []prime = new boolean[nodes + 1];
    Arrays.fill(prime, true);
   
    SieveOfEratosthenes(prime, nodes + 1);
   
    // Hash-table to mark the
    // degree of every node
    int []degree = new int[n + 2 + 1];
   
    // Initially let all the degrees be 1
    for (int i = 1; i <= nodes; i++)
        degree[i] = 1;
   
    // Increase the count of the degree
    for (int i = 0; i < n; i++)
        degree[prufer[i]]++;
   
    // Print the nodes with prime degree
    for (int i = 1; i <= nodes; i++) {
        if (prime[degree[i]]) {
            System.out.print(i+ " ");
        }
    }
}
   
// Driver Code
public static void main(String[] args)
{
    int a[] = { 4, 1, 3, 4 };
    int n = a.length;
   
    PrimeDegreeNodes(a, n);
   
}
}
  
// This code contributed by Princi Singh

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Python3

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# Python3 implementation to print the 
# nodes with prime degree from the
# given prufer sequence
  
# Function to create Sieve
# to check primes
def SieveOfEratosthenes(prime, p_size):
      
    # False here indicates
    # that it is not prime
    prime[0] = False
    prime[1] = False
    p = 2
    while (p * p <= p_size): 
          
        # If prime[p] is not changed, 
        # then it is a prime 
        if (prime[p]): 
              
            # Update all multiples of p,
            # set them to non-prime
            for i in range(p * 2, p_size + 1, p): 
                prime[i] = False
        p += 1
                  
# Function to print the nodes with
# prime degree in the tree
# whose Prufer sequence is given
def PrimeDegreeNodes(prufer, n):
      
    nodes = n + 2
    prime = [True] * (nodes + 1)
    SieveOfEratosthenes(prime, nodes + 1)
      
    # Hash-table to mark the
    # degree of every node
    degree = [0] * (n + 2 + 1);
  
    # Initially let all the degrees be 1
    for i in range(1, nodes + 1):
        degree[i] = 1;
  
    # Increase the count of the degree
    for i in range(0, n):
        degree[prufer[i]] += 1
  
    # Print the nodes with prime degree
    for i in range(1, nodes + 1):
        if prime[degree[i]]:
            print(i, end = ' ')
  
# Driver Code
if __name__=='__main__':
      
    a = [ 4, 1, 3, 4 ]
    n = len(a)
      
    PrimeDegreeNodes(a, n)
  
# This code is contributed by rutvik_56

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C#

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// C# implementation to print the 
// nodes with prime degree from the
// given prufer sequence
using System;
  
class GFG{
  
// Function to create Sieve
// to check primes
static void SieveOfEratosthenes(bool []prime,
                                int p_size)
{
  
    // False here indicates
    // that it is not prime
    prime[0] = false;
    prime[1] = false;
  
    for(int p = 2; p * p <= p_size; p++)
    {
          
       // If prime[p] is not changed,
       // then it is a prime
       if (prime[p]) 
       {
             
           // Update all multiples of p,
           // set them to non-prime
           for(int i = p * 2; i <= p_size; 
                                    i += p)
              prime[i] = false;
        }
    }
}
  
// Function to print the nodes with
// prime degree in the tree
// whose Prufer sequence is given
static void PrimeDegreeNodes(int []prufer, int n)
{
    int nodes = n + 2;
    bool []prime = new bool[nodes + 1];
      
    for(int i = 0; i < prime.Length; i++)
       prime[i] = true;
  
    SieveOfEratosthenes(prime, nodes + 1);
  
    // Hash-table to mark the
    // degree of every node
    int []degree = new int[n + 2 + 1];
  
    // Initially let all the degrees be 1
    for(int i = 1; i <= nodes; i++)
       degree[i] = 1;
  
    // Increase the count of the degree
    for(int i = 0; i < n; i++)
       degree[prufer[i]]++;
  
    // Print the nodes with prime degree
    for(int i = 1; i <= nodes; i++)
    {
       if (prime[degree[i]])
       {
           Console.Write(i + " ");
       }
    }
}
  
// Driver Code
public static void Main(String[] args)
{
    int []a = { 4, 1, 3, 4 };
    int n = a.Length;
  
    PrimeDegreeNodes(a, n);
}
}
  
// This code is contributed by 29AjayKumar

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Output:

1 3 4

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