Maximum edge removal from tree to make even forest

Given an undirected tree which has even number of vertices, we need to remove the maximum number of edges from this tree such that each connected component of the resultant forest has an even number of vertices.
Examples:



In above shown tree, we can remove at max 2 
edges 0-2 and 0-4 shown in red such that each
connected component will have even number of
vertices.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

As we need connected components that have even number of vertices so when we get one component we can remove the edge that connects it to the remaining tree and we will be left with a tree with even number of vertices which will be the same problem but of smaller size, we have to repeat this algorithm until the remaining tree cannot be decomposed further in the above manner.
We will traverse the tree using DFS which will return the number of vertices in the component of which the current node is the root. If a node gets an even number of vertices from one of its children then the edge from that node to its child will be removed and result will be increased by one and if the returned number is odd then we will add it to the number of vertices that the component will have if the current node is the root of it.

1) Do DFS from any starting node as tree
is connected.

2) Initialize count of nodes in subtree rooted
under current node as 0.

3) Do following recursively for every subtree
of current node.
    a) If size of current subtree is even,
       increment result by 1 as we can 
       disconnect the subtree.
    b) Else add count of nodes in current 
       subtree to current count.

Please see below code for better understanding,

C++

/* Program to get maximum number of edges which 
   can be removed such that each connected component 
   of this tree will have an even number of vertices */
#include <bits/stdc++.h> 
using namespace std; 
  
// Utility method to do DFS of the graph and count edge 
// deletion for even forest 
int dfs(vector<int> g[], int u, bool visit[], int& res) 
{ 
    visit[u] = true; 
    int currComponentNode = 0; 
  
    //  iterate over all neighbor of node u 
    for (int i = 0; i < g[u].size(); i++) 
    { 
        int v = g[u][i]; 
        if (!visit[v]) 
        { 
            // Count the number of nodes in a subtree 
            int subtreeNodeCount = dfs(g, v, visit, res); 
  
            // if returned node count is even, disconnect 
            // the subtree and increase result by one. 
            if (subtreeNodeCount % 2 == 0) 
                res++; 
  
            //  else add subtree nodes in current component 
            else
                currComponentNode += subtreeNodeCount; 
        } 
    } 
  
    // number of nodes in current component and one for 
    // current node 
    return (currComponentNode + 1); 
} 
  
/*  method returns max edge that we can remove, after which 
    each  connected component will have even number of 
    vertices */
int maxEdgeRemovalToMakeForestEven(vector<int> g[], int N) 
{ 
    // Create a visited array for DFS and make all nodes 
    // unvisited in starting 
    bool visit[N + 1]; 
    for (int i = 0; i <= N; i++) 
        visit[i] = false; 
  
    int res = 0; // Passed as reference 
  
    //  calling the dfs from node-0 
    dfs(g, 0, visit, res); 
  
    return res; 
} 
  
// Utility function to add an undirected edge (u,v) 
void addEdge(vector<int> g[], int u, int v) 
{ 
    g[u].push_back(v); 
    g[v].push_back(u); 
} 
  
//  Driver code to test above methods 
int main() 
{ 
    int edges[][2] = {{0, 2}, {0, 1}, {0, 4}, 
                      {2, 3}, {4, 5}, {5, 6}, 
                      {5, 7}}; 
    int N = sizeof(edges)/sizeof(edges[0]); 
    vector<int> g[N + 1]; 
    for (int i = 0; i < N; i++) 
         addEdge(g, edges[i][0], edges[i][1]); 
  
    cout << maxEdgeRemovalToMakeForestEven(g, N); 
    return 0; 
} 

Java

/* Program to get maximum number of edges which  
can be removed such that each connected component  
of this tree will have an even number of vertices */
import java.util.*; 
class GFG 
{ 
      
// graph 
static Vector<Vector<Integer>> g = new Vector<Vector<Integer>>(); 
  
static int res; 
  
// Utility method to do DFS of the graph and count edge  
// deletion for even forest  
static int dfs( int u, boolean visit[])  
{  
    visit[u] = true;  
    int currComponentNode = 0;  
  
    // iterate over all neighbor of node u  
    for (int i = 0; i < g.get(u).size(); i++)  
    {  
        int v = g.get(u).get(i);  
        if (!visit[v])  
        {  
            // Count the number of nodes in a subtree  
            int subtreeNodeCount = dfs(v, visit);  
  
            // if returned node count is even, disconnect  
            // the subtree and increase result by one.  
            if (subtreeNodeCount % 2 == 0)  
                res++;  
  
            // else add subtree nodes in current component  
            else
                currComponentNode += subtreeNodeCount;  
        }  
    }  
  
    // number of nodes in current component and one for  
    // current node  
    return (currComponentNode + 1);  
}  
  
/* method returns max edge that we can remove, after which  
    each connected component will have even number of  
    vertices */
static int maxEdgeRemovalToMakeForestEven( int N)  
{  
    // Create a visited array for DFS and make all nodes  
    // unvisited in starting  
    boolean visit[]=new boolean[N + 1];  
    for (int i = 0; i <= N; i++)  
        visit[i] = false;  
  
    res = 0; // Passed as reference  
  
    // calling the dfs from node-0  
    dfs(0, visit);  
  
    return res;  
}  
  
// Utility function to add an undirected edge (u,v)  
static void addEdge( int u, int v)  
{  
    g.get(u).add(v);  
    g.get(v).add(u);  
}  
  
// Driver code to test above methods  
public static void main(String args[])  
{  
    int edges[][] = {{0, 2}, {0, 1}, {0, 4},  
                    {2, 3}, {4, 5}, {5, 6},  
                    {5, 7}};  
    int N = edges.length;  
      
    for (int i = 0; i < N+1; i++) 
    g.add(new Vector<Integer>()); 
      
    for (int i = 0; i < N; i++)  
        addEdge( edges[i][0], edges[i][1]);  
          
  
    System.out.println(maxEdgeRemovalToMakeForestEven( N));  
} 
} 
  
// This code is contributed by Arnab Kundu 

C#

/* C# Program to get maximum number of edges which  
can be removed such that each connected component  
of this tree will have an even number of vertices */
using System; 
using System.Collections.Generic; 
  
class GFG 
{ 
      
// graph 
static List<List<int>> g = new List<List<int>>(); 
  
static int res; 
  
// Utility method to do DFS of the graph and  
// count edge deletion for even forest  
static int dfs( int u, bool []visit)  
{  
    visit[u] = true;  
    int currComponentNode = 0;  
  
    // iterate over all neighbor of node u  
    for (int i = 0; i < g[u].Count; i++)  
    {  
        int v = g[u][i];  
        if (!visit[v])  
        {  
            // Count the number of nodes in a subtree  
            int subtreeNodeCount = dfs(v, visit);  
  
            // if returned node count is even, disconnect  
            // the subtree and increase result by one.  
            if (subtreeNodeCount % 2 == 0)  
                res++;  
  
            // else add subtree nodes in current component  
            else
                currComponentNode += subtreeNodeCount;  
        }  
    }  
  
    // number of nodes in current component and one for  
    // current node  
    return (currComponentNode + 1);  
}  
  
/* method returns max edge that we can remove, 
after which each connected component  
will have even number of vertices */
static int maxEdgeRemovalToMakeForestEven( int N)  
{  
    // Create a visited array for DFS and  
    // make all nodes unvisited in starting  
    bool []visit = new bool[N + 1];  
    for (int i = 0; i <= N; i++)  
        visit[i] = false;  
  
    res = 0; // Passed as reference  
  
    // calling the dfs from node-0  
    dfs(0, visit);  
  
    return res;  
}  
  
// Utility function to add an undirected edge (u,v)  
static void addEdge( int u, int v)  
{  
    g[u].Add(v);  
    g[v].Add(u);  
}  
  
// Driver code 
public static void Main(String []args)  
{  
    int [,]edges = {{0, 2}, {0, 1}, {0, 4},  
                    {2, 3}, {4, 5}, {5, 6},  
                    {5, 7}};  
    int N = edges.GetLength(0);  
      
    for (int i = 0; i < N + 1; i++) 
    g.Add(new List<int>()); 
      
    for (int i = 0; i < N; i++)  
        addEdge(edges[i, 0], edges[i, 1]);  
  
    Console.WriteLine(maxEdgeRemovalToMakeForestEven(N));  
} 
} 
  
// This code is contributed by 29AjayKumar 

Output:

Time Complexity : O(n) where n is number of nodes in tree.

This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

C#

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