Euler Tour of Tree (original) (raw)

Last Updated : 12 Jan, 2023

A Tree is a generalization of connected graph where it has N nodes that will have exactly N-1 edges, i.e one edge between every pair of vertices. Find the Euler tour of tree represented by adjacency list.

Examples:

Input :

Output : 1 2 3 2 4 2 1

Input :

Output : 1 5 4 2 4 3 4 5 1

Euler tour is defined as a way of traversing tree such that each vertex is added to the tour when we visit it (either moving down from parent vertex or returning from child vertex). We start from root and reach back to root after visiting all vertices.
It requires exactly 2*N-1 vertices to store Euler tour.

Approach: We will run DFS(Depth first search) algorithm on Tree as:

  1. Visit root node, i.e 1
    vis[1]=1, Euler[0]=1
    run dfs() for all unvisited adjacent nodes(2)
  2. Visit node 2
    vis[2]=1, Euler[1]=2
    run dfs() for all unvisited adjacent nodes(3, 4)
  3. Visit node 3
    vis[3]=1, Euler[2]=3
    All adjacent nodes are already visited, return to parent node
    and add parent to Euler tour Euler[3]=2
  4. Visit node 4
    vis[4]=1, Euler[4]=4
    All adjacent nodes are already visited, return to parent node
    and add parent to Euler tour, Euler[5]=2
  5. Visit node 2
    All adjacent nodes are already visited, return to parent node
    and add parent to Euler tour, Euler[6]=1
  6. Visit node 1
    All adjacent nodes are already visited, and node 1 is root node
    so, we stop our recursion here.

Similarly, for example 2:

Implementation:

C++ `

// C++ program to print Euler tour of a // tree. #include <bits/stdc++.h> using namespace std;

#define MAX 1001

// Adjacency list representation of tree vector adj[MAX];

// Visited array to keep track visited // nodes on tour int vis[MAX];

// Array to store Euler Tour int Euler[2 * MAX];

// Function to add edges to tree void add_edge(int u, int v) { adj[u].push_back(v); adj[v].push_back(u); }

// Function to store Euler Tour of tree void eulerTree(int u, int &index) { vis[u] = 1; Euler[index++] = u; for (auto it : adj[u]) { if (!vis[it]) { eulerTree(it, index); Euler[index++] = u; } } }

// Function to print Euler Tour of tree void printEulerTour(int root, int N) { int index = 0;
eulerTree(root, index); for (int i = 0; i < (2*N-1); i++) cout << Euler[i] << " "; }

// Driver code int main() { int N = 4;

add_edge(1, 2);
add_edge(2, 3);
add_edge(2, 4);

// Consider 1 as root and print
// Euler tour 
printEulerTour(1, N);

return 0;

}

Java

// Java program to print Euler tour of a // tree. import java.util.*;

class GFG{

static final int MAX = 1001; static int index = 0;

// Adjacency list representation of tree static ArrayList< ArrayList> adj = new ArrayList<>();

// Visited array to keep track visited // nodes on tour static int vis[] = new int[MAX];

// Array to store Euler Tour static int Euler[] = new int[2 * MAX];

// Function to add edges to tree static void add_edge(int u, int v) { adj.get(u).add(v); adj.get(v).add(u); }

// Function to store Euler Tour of tree static void eulerTree(int u) { vis[u] = 1; Euler[index++] = u;

for(int it : adj.get(u))
{
    if (vis[it] == 0) 
    {
        eulerTree(it);
        Euler[index++] = u;
    }
}

}

// Function to print Euler Tour of tree static void printEulerTour(int root, int N) { eulerTree(root); for(int i = 0; i < (2 * N - 1); i++) System.out.print(Euler[i] + " "); }

// Driver code public static void main(String[] args) { int N = 4;

for(int i = 0; i <= N; i++)
    adj.add(new ArrayList<>());
    
add_edge(1, 2);
add_edge(2, 3);
add_edge(2, 4);

// Consider 1 as root and print
// Euler tour
printEulerTour(1, N);

} }

// This code is contributed by jrishabh99

Python3

Python program to print Euler tour of a tree.

from collections import defaultdict

Adjacency list representation of tree

adj = defaultdict(list)

Visited dictionary to keep track of visited nodes on our tour

vis = defaultdict(bool)

defaultdict to store Euler Tour

MAX = 1001 Euler = [0](2MAX)

Function to add edges to tree

def add_edge(u, v): adj[u].append(v) adj[v].append(u)

Function to store Euler Tour of Tree

def eulerTree(u, index): vis[u] = True Euler[index] = u index += 1 for nbr in adj[u]: if not vis[nbr]: index = eulerTree(nbr, index) Euler[index] = u index += 1 return index

Function to print Euler Tour of Tree

def printEulerTour(root, N): index = 0 eulerTree(root, index) for i in range(2*N-1): print(Euler[i], end=" ")

Driver Code

N = 4 add_edge(1, 2) add_edge(2, 3) add_edge(2, 4)

printEulerTour(1, N)

C#

// C# code for the above approach using System; using System.Collections.Generic;

class GFG { const int MAX = 1001; static int index = 0;

// Adjacency list representation of tree
static List<List<int>> adj = new List<List<int>>();

// Visited array to keep track visited
// nodes on tour
static int[] vis = new int[MAX];

// Array to store Euler Tour
static int[] Euler = new int[2 * MAX];

// Function to add edges to tree
static void add_edge(int u, int v)
{
    adj[u].Add(v);
    adj[v].Add(u);
}

// Function to store Euler Tour of tree
static void eulerTree(int u)
{
    vis[u] = 1;
    Euler[index++] = u;

    foreach (int it in adj[u])
    {
        if (vis[it] == 0)
        {
            eulerTree(it);
            Euler[index++] = u;
        }
    }
}

// Function to print Euler Tour of tree
static void printEulerTour(int root, int N)
{
    eulerTree(root);
    for (int i = 0; i < (2 * N - 1); i++)
        Console.Write(Euler[i] + " ");
}

// Driver code
public static void Main(string[] args)
{
    int N = 4;

    for (int i = 0; i <= N; i++)
        adj.Add(new List<int>());

    add_edge(1, 2);
    add_edge(2, 3);
    add_edge(2, 4);

    // Consider 1 as root and print
    // Euler tour
    printEulerTour(1, N);
}

}

// This code is contributed by lokeshpotta20.

JavaScript

`

Complexity Analysis: