Random Tree Generator Using Prüfer Sequence with Examples

Given an integer N, the task is to generate a random labelled tree of N node with (N – 1) edges without forming cycle.

Note: The output generated below is random which may not match with the output generated by the code.

Examples:

Input: N = 3
Output:
1 3
1 2

Input: N = 5
Output:
3 2
4 3
1 4
1 5

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

This approach uses the Prüfer Sequence to generate random trees.

What is a Prüfer Sequence?
In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labelled tree is a unique sequence associated with the tree. The sequence for a tree on n vertices has length n – 2 and can be generated by a simple iterative algorithm.

If the number of nodes is N then the Prüfer Sequence is of length (N – 2) and each position can have N possible values. So the number of the possible labeled trees with N Nodes is N^{(N – 2)}.

How Random Trees are generated using Prüfer Sequence?
Generally, Random Tree Generation with N nodes is done in the following steps:

Generate a Random Sequence
```
S = {s₁, s₂, s₃.....s_n-2}
```
where each element of the sequence s_i ∈ {1, 2, 3, … N} where repetition of elements is allowed
Generate Tree from the generated Prüfer Sequence S:
1. Create N nodes with values {1, 2, 3, … N}
2. Find smallest element X such that X ∈ {1, 2, 3, … N} and X ∉ S
3. Join Node with value X to the node with value s₁
4. Delete s₁ from S
5. Repeat the same process from step 2 with untill the Prüfer Sequence is empty.

For Example:

Number of Nodes = 3
Then the Prüfer Sequence will be of length (N – 2), as in this case it will be of 1 and the possible values it can have {1, 2, 3}.
Possible Random Sequences will be {{1}, {2}, {3}}.

Below is the implementation of the above approach.

C++

// C++ Implementation for random 
// tree generator using Prufer Sequence 
#include<bits/stdc++.h> 
using namespace std; 
  
// Prints edges of tree 
// represented by give Prufer code 
void printTreeEdges(int prufer[], int m) 
{ 
    int vertices = m + 2; 
    int vertex_set[vertices]; 
  
    // Initialize the array of vertices 
    for (int i = 0; i < vertices; i++) 
        vertex_set[i] = 0; 
  
    // Number of occurrences of vertex in code 
    for (int i = 0; i < vertices - 2; i++) 
        vertex_set[prufer[i] - 1] += 1; 
  
    cout<<("\nThe edge set E(G) is:\n"); 
  
    int j = 0; 
  
    // Find the smallest label not present in 
    // prufer[]. 
    for (int i = 0; i < vertices - 2; i++)  
    { 
        for (j = 0; j < vertices; j++) 
        { 
  
            // If j+1 is not present in prufer set 
            if (vertex_set[j] == 0) 
            { 
  
                // Remove from Prufer set and print 
                // pair. 
                vertex_set[j] = -1; 
                cout<<"(" << (j + 1) << ", "
                                << prufer[i] << ") "; 
  
                vertex_set[prufer[i] - 1]--; 
  
                break; 
            } 
        } 
    } 
  
    j = 0; 
  
    // For the last element 
    for (int i = 0; i < vertices; i++) 
    { 
        if (vertex_set[i] == 0 && j == 0) 
        { 
  
            cout << "(" << (i + 1) << ", "; 
            j++; 
        } 
        else if (vertex_set[i] == 0 && j == 1) 
            cout << (i + 1) << ")\n"; 
    } 
} 
  
// generate random numbers in between l an r 
int ran(int l, int r) 
{ 
    return l + (rand() % (r - l + 1)); 
} 
  
// Function to Generate Random Tree 
void generateRandomTree(int n) 
{ 
  
    int length = n - 2; 
    int arr[length]; 
  
    // Loop to Generate Random Array 
    for (int i = 0; i < length; i++)  
    { 
        arr[i] = ran(0, pow(2, length + 1)) + 1; 
    } 
    printTreeEdges(arr, length); 
} 
  
// Driver Code 
int main() 
{ 
    srand(time(0)); 
    int n = 5; 
    generateRandomTree(n); 
  
    return 0; 
} 
  
// This code is contributed by Arnab Kundu 

Java

// Java Implementation for random 
// tree generator using Prufer Sequence 
  
import java.util.Arrays; 
import java.util.Random; 
  
class GFG { 
  
    // Prints edges of tree 
    // represented by give Prufer code 
    static void printTreeEdges(int prufer[], int m) 
    { 
        int vertices = m + 2; 
        int vertex_set[] = new int[vertices]; 
  
        // Initialize the array of vertices 
        for (int i = 0; i < vertices; i++) 
            vertex_set[i] = 0; 
  
        // Number of occurrences of vertex in code 
        for (int i = 0; i < vertices - 2; i++) 
            vertex_set[prufer[i] - 1] += 1; 
  
        System.out.print("\nThe edge set E(G) is:\n"); 
  
        int j = 0; 
  
        // Find the smallest label not present in 
        // prufer[]. 
        for (int i = 0; i < vertices - 2; i++) { 
            for (j = 0; j < vertices; j++) { 
  
                // If j+1 is not present in prufer set 
                if (vertex_set[j] == 0) { 
  
                    // Remove from Prufer set and print 
                    // pair. 
                    vertex_set[j] = -1; 
                    System.out.print("(" + (j + 1) + ", "
                                     + prufer[i] + ") "); 
  
                    vertex_set[prufer[i] - 1]--; 
  
                    break; 
                } 
            } 
        } 
  
        j = 0; 
  
        // For the last element 
        for (int i = 0; i < vertices; i++) { 
            if (vertex_set[i] == 0 && j == 0) { 
  
                System.out.print("(" + (i + 1) + ", "); 
                j++; 
            } 
            else if (vertex_set[i] == 0 && j == 1) 
                System.out.print((i + 1) + ")\n"); 
        } 
    } 
  
    // Function to Generate Random Tree 
    static void generateRandomTree(int n) 
    { 
  
        Random rand = new Random(); 
        int length = n - 2; 
        int[] arr = new int[length]; 
  
        // Loop to Generate Random Array 
        for (int i = 0; i < length; i++) { 
            arr[i] = rand.nextInt(length + 1) + 1; 
        } 
        printTreeEdges(arr, length); 
    } 
  
    // Driver Code 
    public static void main(String[] args) 
    { 
        int n = 5; 
        generateRandomTree(n); 
    } 
} 

C#

// C# Implementation for random 
// tree generator using Prufer Sequence 
using System; 
  
class GFG 
{ 
  
    // Prints edges of tree 
    // represented by give Prufer code 
    static void printTreeEdges(int []prufer, int m) 
    { 
        int vertices = m + 2; 
        int []vertex_set = new int[vertices]; 
  
        // Initialize the array of vertices 
        for (int i = 0; i < vertices; i++) 
            vertex_set[i] = 0; 
  
        // Number of occurrences of vertex in code 
        for (int i = 0; i < vertices - 2; i++) 
            vertex_set[prufer[i] - 1] += 1; 
  
        Console.Write("\nThe edge set E(G) is:\n"); 
  
        int j = 0; 
  
        // Find the smallest label not present in 
        // prufer[]. 
        for (int i = 0; i < vertices - 2; i++)  
        { 
            for (j = 0; j < vertices; j++) 
            { 
  
                // If j + 1 is not present in prufer set 
                if (vertex_set[j] == 0) 
                { 
  
                    // Remove from Prufer set and print 
                    // pair. 
                    vertex_set[j] = -1; 
                    Console.Write("(" + (j + 1) + ", "
                                    + prufer[i] + ") "); 
  
                    vertex_set[prufer[i] - 1]--; 
  
                    break; 
                } 
            } 
        } 
  
        j = 0; 
  
        // For the last element 
        for (int i = 0; i < vertices; i++) 
        { 
            if (vertex_set[i] == 0 && j == 0)  
            { 
  
                Console.Write("(" + (i + 1) + ", "); 
                j++; 
            } 
            else if (vertex_set[i] == 0 && j == 1) 
                Console.Write((i + 1) + ")\n"); 
        } 
    } 
  
    // Function to Generate Random Tree 
    static void generateRandomTree(int n) 
    { 
  
        Random rand = new Random(); 
        int length = n - 2; 
        int[] arr = new int[length]; 
  
        // Loop to Generate Random Array 
        for (int i = 0; i < length; i++)  
        { 
            arr[i] = rand.Next(length + 1) + 1; 
        } 
        printTreeEdges(arr, length); 
    } 
  
    // Driver Code 
    public static void Main(String[] args) 
    { 
        int n = 5; 
        generateRandomTree(n); 
    } 
} 
  
// This code is contributed by 29AjayKumar 

Output:

The edge set E(G) is:
(2, 4) (4, 3) (3, 1) (1, 5)

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

C#

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