Size of the Largest Trees in a Forest formed by the given Graph

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Given an undirected acyclic graph having N nodes and M edges, the task is to find the size of the largest tree in the forest formed by the graph.

A forest is a collection of disjoint trees. In other words, we can also say that forest is a collection of an acyclic graph which is not connected.

Examples:

Input: N = 5, edges[][] = {{0, 1}, {0, 2}, {3, 4}}
Output: 3
Explanation:
There are 2 trees, each having size 3 and 2 respectively.
   0
 /   \
1     2
and
3
 \
  4
Hence the size of the largest tree is 3.

Input: N = 5, edges[][] = {{0, 1}, {0, 2}, {3, 4}, {0, 4}, {3, 5}}
Output: 6

Approach: The idea is to first count the number of reachable nodes from every forest. Therefore:

Apply DFS on every node and obtain the size of the tree formed by this node and check if every connected node is visited from one source.
If the size of the current tree is greater than the answer then update the answer to the current tree’s size.
Again perform DFS traversal if some set of nodes are not yet visited.
Finally, the max of all the answers when all the nodes are visited is the final answer.

Below is the implementation of the above approach:

C++

// C++ program to find the size
// of the largest tree in the forest
 
#include <bits/stdc++.h>
using namespace std;
 
// A utility function to add
// an edge in an undirected graph.
void addEdge(vector<int> adj[],
             int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}
 
// A utility function to perform DFS of a
// graph recursively from a given vertex u
// and returns the size of the tree formed by u
int DFSUtil(int u, vector<int> adj[],
            vector<bool>& visited)
{
    visited[u] = true;
    int sz = 1;
 
    // Iterating through all the nodes
    for (int i = 0; i < adj[u].size(); i++)
        if (visited[adj[u][i]] == false)
 
            // Perform DFS if the node is
            // not yet visited
            sz += DFSUtil(
                adj[u][i], adj, visited);
    return sz;
}
 
// Function to return the  size of the
// largest tree in the forest given as
// the adjacency list
int largestTree(vector<int> adj[], int V)
{
    vector<bool> visited(V, false);
    int answer = 0;
 
    // Iterating through all the vertices
    for (int u = 0; u < V; u++) {
        if (visited[u] == false) {
 
            // Find the answer
            answer
                = max(answer,
                      DFSUtil(u, adj, visited));
        }
    }
    return answer;
}
 
// Driver code
int main()
{
    int V = 5;
    vector<int> adj[V];
    addEdge(adj, 0, 1);
    addEdge(adj, 0, 2);
    addEdge(adj, 3, 4);
    cout << largestTree(adj, V);
    return 0;
}

Java

// Java program to find the size
// of the largest tree in the forest
import java.util.*;
class GFG{
 
// A utility function to add
// an edge in an undirected graph.
static void addEdge(Vector<Integer> adj[],
                    int u, int v)
{
  adj[u].add(v);
  adj[v].add(u);
}
 
// A utility function to perform DFS of a
// graph recursively from a given vertex u
// and returns the size of the tree formed by u
static int DFSUtil(int u, Vector<Integer> adj[],
                   Vector<Boolean> visited)
{
  visited.add(u, true);
  int sz = 1;
 
  // Iterating through all the nodes
  for (int i = 0; i < adj[u].size(); i++)
    if (visited.get(adj[u].get(i)) == false)
 
      // Perform DFS if the node is
      // not yet visited
      sz += DFSUtil(adj[u].get(i), 
                    adj, visited);
  return sz;
}
 
// Function to return the  size of the
// largest tree in the forest given as
// the adjacency list
static int largestTree(Vector<Integer> adj[], 
                       int V)
{
  Vector<Boolean> visited = new Vector<>();
  for(int i = 0; i < V; i++) 
  {
    visited.add(false);
  }
  int answer = 0;
 
  // Iterating through all the vertices
  for (int u = 0; u < V; u++) 
  {
    if (visited.get(u) == false) 
    {
      // Find the answer
      answer = Math.max(answer, 
               DFSUtil(u, adj, visited));
    }
  }
  return answer;
}
 
// Driver code
public static void main(String[] args)
{
  int V = 5;
  Vector<Integer> adj[] = new Vector[V];
  for (int i = 0; i < adj.length; i++)
    adj[i] = new Vector<Integer>();
  addEdge(adj, 0, 1);
  addEdge(adj, 0, 2);
  addEdge(adj, 3, 4);
  System.out.print(largestTree(adj, V));
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 program to find the size
# of the largest tree in the forest
  
# A utility function to add
# an edge in an undirected graph.
def addEdge(adj, u, v):
 
    adj[u].append(v)
    adj[v].append(u)
 
# A utility function to perform DFS of a
# graph recursively from a given vertex u
# and returns the size of the tree formed by u
def DFSUtil(u, adj, visited):
     
    visited[u] = True
    sz = 1
  
    # Iterating through all the nodes
    for i in range(0, len(adj[u])):
        if (visited[adj[u][i]] == False):
  
            # Perform DFS if the node is
            # not yet visited
            sz += DFSUtil(adj[u][i], adj, visited)
             
    return sz
 
# Function to return the  size of the
# largest tree in the forest given as
# the adjacency list
def largestTree(adj, V):
     
    visited = [False for i in range(V)]
     
    answer = 0
  
    # Iterating through all the vertices
    for u in range(V):
        if (visited[u] == False):
  
            # Find the answer
            answer = max(answer,DFSUtil(
                u, adj, visited))
         
    return answer
 
# Driver code
if __name__=="__main__":
 
    V = 5
     
    adj = [[] for i in range(V)]
     
    addEdge(adj, 0, 1)
    addEdge(adj, 0, 2)
    addEdge(adj, 3, 4)
     
    print(largestTree(adj, V))
     
# This code is contributed by rutvik_56

C#

// C# program to find the size
// of the largest tree in the forest
using System;
using System.Collections.Generic;
class GFG{
 
// A utility function to add
// an edge in an undirected graph.
static void addEdge(List<int> []adj,
                    int u, int v)
{
  adj[u].Add(v);
  adj[v].Add(u);
}
 
// A utility function to perform DFS of a
// graph recursively from a given vertex u
// and returns the size of the tree formed by u
static int DFSUtil(int u, List<int> []adj,
                   List<Boolean> visited)
{
  visited.Insert(u, true);
  int sz = 1;
 
  // Iterating through all the nodes
  for (int i = 0; i < adj[u].Count; i++)
    if (visited[adj[u][i]] == false)
 
      // Perform DFS if the node is
      // not yet visited
      sz += DFSUtil(adj[u][i], 
                    adj, visited);
  return sz;
}
 
// Function to return the  size of the
// largest tree in the forest given as
// the adjacency list
static int largestTree(List<int> []adj, 
                       int V)
{
  List<Boolean> visited = new List<Boolean>();
  for(int i = 0; i < V; i++) 
  {
    visited.Add(false);
  }
  int answer = 0;
 
  // Iterating through all the vertices
  for (int u = 0; u < V; u++) 
  {
    if (visited[u] == false) 
    {
      // Find the answer
      answer = Math.Max(answer, 
               DFSUtil(u, adj, visited));
    }
  }
  return answer;
}
 
// Driver code
public static void Main(String[] args)
{
  int V = 5;
  List<int> []adj = new List<int>[V];
   
  for (int i = 0; i < adj.Length; i++)
    adj[i] = new List<int>();
   
  addEdge(adj, 0, 1);
  addEdge(adj, 0, 2);
  addEdge(adj, 3, 4);
  Console.Write(largestTree(adj, V));
}
}
 
// This code is contributed by Rajput-Ji

Javascript

<script>
    // Javascript program to find the size
    // of the largest tree in the forest
     
    // A utility function to add
    // an edge in an undirected graph.
    function addEdge(adj, u, v)
    {
        adj[u].push(v);
        adj[v].push(u);
    }
 
    // A utility function to perform DFS of a
    // graph recursively from a given vertex u
    // and returns the size of the tree formed by u
    function DFSUtil(u, adj, visited)
    {
        visited[u] = true;
        let sz = 1;
 
        // Iterating through all the nodes
        for(let i = 0; i < adj[u].length; i++)
        {
            if (visited[adj[u][i]] == false)
            {
                // Perform DFS if the node is
                // not yet visited
                sz += DFSUtil(adj[u][i], adj, visited);
            }
        }
 
        return sz;
     }
 
    // Function to return the  size of the
    // largest tree in the forest given as
    // the adjacency list
    function largestTree(adj, V)
    {
        let visited = new Array(V);
        visited.fill(false);
 
        let answer = 0;
 
        // Iterating through all the vertices
        for(let u = 0; u < V; u++)
        {
            if (visited[u] == false)
            {
                // Find the answer
                answer = Math.max(answer,DFSUtil(u, adj, visited));
            }
        }
 
        return answer;
    }
     
    let V = 5;
      
    let adj = [];
    for(let i = 0; i < V; i++)
    {
        adj.push([]);
    }
      
    addEdge(adj, 0, 1);
    addEdge(adj, 0, 2);
    addEdge(adj, 3, 4);
      
    document.write(largestTree(adj, V));
     
    // This code is contributed by divyesh072019.
</script>

Output:

Time Complexity: O(V + E), where V is the number of vertices and E is the number of edges.

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