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

  • Difficulty Level : Hard
  • Last Updated : 24 Sep, 2021

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.

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Examples:  



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

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: 
3

 

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




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