Total nodes traversed in Euler Tour Tree

Euler tour of tree has been already discussed which flattens the hierarchical structure of tree into array which contains exactly 2*N-1 values. In this post, the task is to prove that the degree of Euler Tour Tree is 2 times the number of nodes minus one. Here degree means the total number of nodes get traversed in Euler Tour.

Examples:

Input: 

Output: 15

Input: 

Output: 17 

Explanation: 

Using Example 1:

where

It can be seen that each node’s count in Euler Tour is exactly equal to the out-degree of node plus 1. 
Mathematically, it can be represented as: 

Where,
Total represents total number of nodes in Euler Tour Tree node_i     represents ith node in given Tree @N represents the total number of node in given Tree@Out_D_e_g[node_i]     represents number of child of node_i     

Implementation:

C++

// C++ program to check the number of nodes
// in Euler Tour tree.
#include <bits/stdc++.h>
using namespace std;

#define MAX 1001

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

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

// Program to check if calculated Value is
// equal to 2*size-1
void checkTotalNumberofNodes(int actualAnswer,
                              int size)
{
    int calculatedAnswer = size;

    // Add out-degree of each node
    for (int i = 1; i <= size; i++)
        calculatedAnswer += adj[i].size();

    if (actualAnswer == calculatedAnswer)
        cout << "Calculated Answer is " << calculatedAnswer
                     << " and is Equal to Actual Answer\n";
    else
        cout << "Calculated Answer is Incorrect\n";
}
int main()
{ // Constructing 1st tree from example
    int N = 8;
    add_edge(1, 2);
    add_edge(1, 3);
    add_edge(2, 4);
    add_edge(2, 5);
    add_edge(3, 6);
    add_edge(3, 7);
    add_edge(6, 8);

    // Out_deg[node[i]] is equal to adj[i].size()
    checkTotalNumberofNodes(2 * N - 1, N);

    // clear previous stored tree
    for (int i = 1; i <= N; i++)
        adj[i].clear();

    // Constructing 2nd tree from example
    N = 9;
    add_edge(1, 2);
    add_edge(1, 3);
    add_edge(2, 4);
    add_edge(2, 5);
    add_edge(2, 6);
    add_edge(3, 9);
    add_edge(5, 7);
    add_edge(5, 8);

    // Out_deg[node[i]] is equal to adj[i].size()
    checkTotalNumberofNodes(2 * N - 1, N);

    return 0;
}

Java

// Java program to check the number of nodes
// in Euler Tour tree.
import java.util.*;

class GFG
{
    static final int MAX = 1001;

    // Adjacency list representation of tree
    static Vector<Integer>[] adj = new Vector[MAX];

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

    // Program to check if calculated Value is
    // equal to 2*size-1
    static void checkTotalNumberofNodes(int actualAnswer,
                                        int size)
    {
        int calculatedAnswer = size;

        // Add out-degree of each node
        for (int i = 1; i <= size; i++)
            calculatedAnswer += adj[i].size();

        if (actualAnswer == calculatedAnswer)
            System.out.print("Calculated Answer is " +
                                    calculatedAnswer +
                  " and is Equal to Actual Answer\n");
        else
            System.out.print("Calculated Answer is Incorrect\n");
    }

    // Driver Code
    public static void main(String[] args)
    {
        for (int i = 0; i < MAX; i++)
            adj[i] = new Vector<Integer>();

        // Constructing 1st tree from example
        int N = 8;
        add_edge(1, 2);
        add_edge(1, 3);
        add_edge(2, 4);
        add_edge(2, 5);
        add_edge(3, 6);
        add_edge(3, 7);
        add_edge(6, 8);

        // Out_deg[node[i]] is equal to adj[i].size()
        checkTotalNumberofNodes(2 * N - 1, N);

        // clear previous stored tree
        for (int i = 1; i <= N; i++)
            adj[i].clear();

        // Constructing 2nd tree from example
        N = 9;
        add_edge(1, 2);
        add_edge(1, 3);
        add_edge(2, 4);
        add_edge(2, 5);
        add_edge(2, 6);
        add_edge(3, 9);
        add_edge(5, 7);
        add_edge(5, 8);

        // Out_deg[node[i]] is equal to adj[i].size()
        checkTotalNumberofNodes(2 * N - 1, N);
    }
}

// This code is contributed by Rajput-Ji

Python3

# Python3 program to check the number
# of nodes in Euler Tour tree.
MAX = 1001;

# Adjacency list representation of tree
adj = [list() for _ in range(MAX)]

# Function to add edges to tree
def add_edge(u, v):
    l1 = adj[u]
    l1.append(v)
    adj[u] = l1;

# Program to check if calculated Value is
# equal to 2*size-1
def checkTotalNumberofNodes(actualAnswer, size) :
    calculatedAnswer = size;

    # push out-degree of each node
    for i in range(1, size + 1):
        calculatedAnswer += len(adj[i]);
    if (actualAnswer == calculatedAnswer):
        print("Calculated Answer is", calculatedAnswer, "and is Equal to Actual Answer")
    else:
        print("Calculated Answer is Incorrect");

# Driver Code

# Constructing 1st tree from example
N = 8;
add_edge(1, 2);
add_edge(1, 3);
add_edge(2, 4);
add_edge(2, 5);
add_edge(3, 6);
add_edge(3, 7);
add_edge(6, 8);

# Out_deg[node[i]] is equal to adj[i].Count
checkTotalNumberofNodes(2 * N - 1, N);

# clear previous stored tree
for i in range(N + 1):
    adj[i] = []

# Constructing 2nd tree from example
N = 9;
add_edge(1, 2);
add_edge(1, 3);
add_edge(2, 4);
add_edge(2, 5);
add_edge(2, 6);
add_edge(3, 9);
add_edge(5, 7);
add_edge(5, 8);

# Out_deg[node[i]] is equal to adj[i].Count
checkTotalNumberofNodes(2 * N - 1, N);

# This code is contributed by phasing17

C#

// C# program to check the number
// of nodes in Euler Tour tree.
using System;
using System.Collections.Generic;

class GFG
{
    static readonly int MAX = 1001;

    // Adjacency list representation of tree
    static List<int>[] adj = new List<int>[MAX];

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

    // Program to check if calculated Value is
    // equal to 2*size-1
    static void checkTotalNumberofNodes(int actualAnswer,
                                        int size)
    {
        int calculatedAnswer = size;

        // Add out-degree of each node
        for (int i = 1; i <= size; i++)
            calculatedAnswer += adj[i].Count;

        if (actualAnswer == calculatedAnswer)
            Console.Write("Calculated Answer is " +
                                 calculatedAnswer +
               " and is Equal to Actual Answer\n");
        else
            Console.Write("Calculated Answer " +
                              "is Incorrect\n");
    }

    // Driver Code
    public static void Main(String[] args)
    {
        for (int i = 0; i < MAX; i++)
            adj[i] = new List<int>();

        // Constructing 1st tree from example
        int N = 8;
        add_edge(1, 2);
        add_edge(1, 3);
        add_edge(2, 4);
        add_edge(2, 5);
        add_edge(3, 6);
        add_edge(3, 7);
        add_edge(6, 8);

        // Out_deg[node[i]] is equal to adj[i].Count
        checkTotalNumberofNodes(2 * N - 1, N);

        // clear previous stored tree
        for (int i = 1; i <= N; i++)
            adj[i].Clear();

        // Constructing 2nd tree from example
        N = 9;
        add_edge(1, 2);
        add_edge(1, 3);
        add_edge(2, 4);
        add_edge(2, 5);
        add_edge(2, 6);
        add_edge(3, 9);
        add_edge(5, 7);
        add_edge(5, 8);

        // Out_deg[node[i]] is equal to adj[i].Count
        checkTotalNumberofNodes(2 * N - 1, N);
    }
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

// Javascript program to check the number
// of nodes in Euler Tour tree.
var MAX = 1001;

// Adjacency list representation of tree
var adj = Array.from(Array(MAX), ()=>Array());

// Function to add edges to tree
function add_edge(u, v)
{
    adj[u].push(v);
}

// Program to check if calculated Value is
// equal to 2*size-1
function checkTotalNumberofNodes(actualAnswer, size)
{
    var calculatedAnswer = size;

    // push out-degree of each node
    for (var i = 1; i <= size; i++)
        calculatedAnswer += adj[i].length;
    if (actualAnswer == calculatedAnswer)
        document.write("Calculated Answer is " +
                             calculatedAnswer +
           " and is Equal to Actual Answer<br>");
    else
        document.write("Calculated Answer " +
                          "is Incorrect<br>");
}

// Driver Code
for (var i = 0; i < MAX; i++)
    adj[i] = [];

// Constructing 1st tree from example
var N = 8;
add_edge(1, 2);
add_edge(1, 3);
add_edge(2, 4);
add_edge(2, 5);
add_edge(3, 6);
add_edge(3, 7);
add_edge(6, 8);

// Out_deg[node[i]] is equal to adj[i].Count
checkTotalNumberofNodes(2 * N - 1, N);

// clear previous stored tree
for (var i = 1; i <= N; i++)
    adj[i] = []

// Constructing 2nd tree from example
N = 9;
add_edge(1, 2);
add_edge(1, 3);
add_edge(2, 4);
add_edge(2, 5);
add_edge(2, 6);
add_edge(3, 9);
add_edge(5, 7);
add_edge(5, 8);

// Out_deg[node[i]] is equal to adj[i].Count
checkTotalNumberofNodes(2 * N - 1, N);

// This code is contributed by itsok.
</script>

Output

Calculated Answer is 15 and is Equal to Actual Answer
Calculated Answer is 17 and is Equal to Actual Answer

Time complexity: O(N)
Auxiliary space: O(MAX)