Prufer Code to Tree Creation

What is Prufer Code?
Given a tree (represented as graph, not as a rooted tree) with n labeled nodes with labels from 1 to n, a Prufer code uniquely identifies the tree. The sequence has n-2 values.

How to get Prufer Code of a tree?

1. Initialize Prufer code as empty.
2. Start with a leaf of lowest label say x. Find the vertex connecting it to the rest of tree say y. Remove x from the tree and add y to the Prufer Code
3. Repeat above step 2 until we are left with two nodes.

A tree with labels from 1 to n.
             5 
           /   \      
          1     4 
        /  \
       2    3

PruferCode = {}
The lowest label leaf is 2, we remove it from tree
and add the other vertex (connecting it to the tree)
to Prufer code
Tree now becomes
             5 
           /   \      
          1     4
           \
            3
Prufer Code becomes = {1}

The lowest label leaf is 3, we remove it from tree
and add the other vertex (connecting it to the tree)
to Prufer code
Tree now becomes
             5 
           /   \      
          1     4
Prufer Code becomes = {1, 1}
           
The lowest label leaf is 1, we remove it from tree
and add the other vertex (connecting it to the tree)
to Prufer code
Tree now becomes
             5 
              \      
               4     
Prufer Code becomes = {1, 1, 5}

We have only two nodes left now, so we stop.

How to construct a tree from given Prufer Code?

Input : (4, 1, 3, 4)
Output : Edges of following tree
         2----4----3----1----5
              |
              6

Input : (1, 3, 5)
Output : Edges of following tree
         2----1----3----5----4

Let the length of given Prufer code be m. The idea is to create an empty graph of m+2 vertices. We remove first element from sequence. Let first element of current sequence be x. Then we find the least value which is not present in the given sequence and not yet added to the tree. Let this value be y. We add an edge from x to y and repeat this step.

Let us understand algorithm to construct tree with above first example: 

Input : (4, 1, 3, 4)

Step 1: First we create an empty graph of 6 vertices 
        and get 4 from the sequence. 
Step 2: Out of 1 to 6, the least vertex not in 
        Prufer sequence is 2. 
Step 3: We form an edge between 2 and 4. 
             2----4    1    3    5      6
Step 4: Next in the sequence is 1 and corresponding 
        vertex with least degree is 5 (as 2 has been 
        considered). 
             2----4    1----5    3    6
Step 5: Next in the sequence is 3 and corresponding 
        vertex with least degree is 1 
        (as 1 is now not part of remaining Prufer sequence) 
             2----4    3----1----5    6
Step 6: Next in the sequence is 4 and corresponding vertex
        with least degree is 3 (as 3 has not been considered 
        as is not present further in sequence)
             2----4----3----1----5    6
Step 7: Finally two vertices are left out from 1 to 6 (4
         and 6) so we join them.
             2----4----3----1----5
                  |
                  6
This is the required tree on 6 vertices.

Following is the implementation.

C++

// C++ program to construct tree from given Prufer Code
#include <bits/stdc++.h>
using namespace std;

// Prints edges of tree represented by given 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";

    // Find the smallest label not present in
    // prufer[].
    int j = 0;
    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";
    }
}

// Driver code
int main()
{
    int prufer[] = { 4, 1, 3, 4 };
    int n = sizeof(prufer) / sizeof(prufer[0]);
    printTreeEdges(prufer, n);
    return 0;
}

Java

// Java program to construct tree from given Prufer Code
class GFG {

    // Prints edges of tree represented by given 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");

        // Find the smallest label not present in
        // prufer[].
        int j = 0;
        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");
        }
    }

    // Driver code
    public static void main(String args[])
    {
        int prufer[] = { 4, 1, 3, 4 };
        int n = prufer.length;
        printTreeEdges(prufer, n);
    }
}

// This code is contributed by Arnab Kundu

Python3

# Python3 program to construct
# tree from given Prufer Code

# Prints edges of tree represented
# by given Prufer code
def printTreeEdges(prufer, m):

    vertices = m + 2

    # Initialize the array of vertices
    vertex_set = [0] * vertices

    # Number of occurrences of vertex in code
    for i in range(vertices - 2):
        vertex_set[prufer[i] - 1] += 1

    print("The edge set E(G) is :")

    # Find the smallest label not present in
    # prufer.
    j = 0
    for i in range(vertices - 2):
        for j in range(vertices):

            # 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
                print("(" , (j + 1),", ",prufer[i],") ",sep = "",end = "")
                vertex_set[prufer[i] - 1] -= 1
                break

    j = 0

    # For the last element
    for i in range(vertices):
        if (vertex_set[i] == 0 and j == 0):
            print("(", (i + 1),", ", sep="", end="")
            j += 1
        else if (vertex_set[i] == 0 and j == 1):
            print((i + 1),")")

# Driver code
prufer = [4, 1, 3, 4]
n = len(prufer)
printTreeEdges(prufer, n)

# This code is contributed by SHUBHAMSINGH10

C#

// C# program to construct tree from given Prufer Code
using System;

class GFG {

    // Prints edges of tree represented by given 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");

        // Find the smallest label not present in
        // prufer[].
        int j = 0;
        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");
        }
    }

    // Driver code
    public static void Main(String[] args)
    {
        int[] prufer = { 4, 1, 3, 4 };
        int n = prufer.Length;
        printTreeEdges(prufer, n);
    }
}

// This code has been contributed by 29AjayKumar

Javascript

<script>

// JavaScript program to construct tree from given Prufer Code

// Prints edges of tree represented by given Prufer code
function printTreeEdges(prufer,m)
{
    let vertices = m + 2;
        let vertex_set = new Array(vertices);

        // Initialize the array of vertices
        for (let i = 0; i < vertices; i++)
            vertex_set[i] = 0;

        // Number of occurrences of vertex in code
        for (let i = 0; i < vertices - 2; i++)
            vertex_set[prufer[i] - 1] += 1;

        document.write("<br>The edge set E(G) is :<br>");

        // Find the smallest label not present in
        // prufer[].
        let j = 0;
        for (let 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;
                    document.write("(" + (j + 1) + ", "
                                     + prufer[i] + ") ");

                    vertex_set[prufer[i] - 1]--;

                    break;
                }
            }
        }

        j = 0;
        // For the last element
        for (let i = 0; i < vertices; i++) {
            if (vertex_set[i] == 0 && j == 0) {
                document.write("(" + (i + 1) + ", ");
                j++;
            }
            else if (vertex_set[i] == 0 && j == 1)
                document.write((i + 1) + ")\n");
        }
}

// Driver code
let prufer=[4, 1, 3, 4];
let n = prufer.length;
printTreeEdges(prufer, n);


// This code is contributed by rag2127

</script>

Output

The edge set E(G) is :
(2, 4)  (5, 1)  (1, 3)  (3, 4)  (4, 6)

Time Complexity: O(n²) where n is the number of vertices in the tree.
Auxiliary Space: O(n)