Diameter of an N-ary tree

The diameter of an N-ary tree is the longest path present between any two nodes of the tree. These two nodes must be two leaf nodes. The following examples have the longest path[diameter] shaded.

Example 1: 
 

diameternary

Example 2: 
 

diametern2



 

Prerequisite: Diameter of a binary tree.
 
The path can either start from one of the nodes and goes up to one of the LCAs of these nodes and again come down to the deepest node of some other subtree or can exist as a diameter of one of the child of the current node. 
The solution will exist in any one of these: 
I] Diameter of one of the children of the current node 
II] Sum of Height of the highest two subtree + 1 
 

C++

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// C++ program to find the height of an N-ary
// tree
#include <bits/stdc++.h>
using namespace std;
 
// Structure of a node of an n-ary tree
struct Node
{
    char key;
    vector<Node *> child;
};
 
// Utility function to create a new tree node
Node *newNode(int key)
{
    Node *temp = new Node;
    temp->key = key;
    return temp;
}
 
// Utility function that will return the depth
// of the tree
int depthOfTree(struct Node *ptr)
{
    // Base case
    if (!ptr)
        return 0;
 
    int maxdepth = 0;
 
    // Check for all children and find
    // the maximum depth
    for (vector<Node*>::iterator it = ptr->child.begin();
                           it != ptr->child.end(); it++)
 
        maxdepth = max(maxdepth , depthOfTree(*it));
 
    return maxdepth + 1;
}
 
// Function to calculate the diameter
// of the tree
int diameter(struct Node *ptr)
{
    // Base case
    if (!ptr)
        return 0;
 
    // Find top two highest children
    int max1 = 0, max2 = 0;
    for (vector<Node*>::iterator it = ptr->child.begin();
                          it != ptr->child.end(); it++)
    {
        int h = depthOfTree(*it);
        if (h > max1)
           max2 = max1, max1 = h;
        else if (h > max2)
           max2 = h;
    }
 
    // Iterate over each child for diameter
    int maxChildDia = 0;
    for (vector<Node*>::iterator it = ptr->child.begin();
                           it != ptr->child.end(); it++)
        maxChildDia = max(maxChildDia, diameter(*it));
 
    return max(maxChildDia, max1 + max2 + 1);
}
 
// Driver program
int main()
{
    /*   Let us create below tree
    *           A
    *         / /  \  \
    *       B  F   D  E
    *      / \     |  /|\
    *     K  J    G  C H I
    *      /\            \
    *    N   M            L
    */
 
    Node *root = newNode('A');
    (root->child).push_back(newNode('B'));
    (root->child).push_back(newNode('F'));
    (root->child).push_back(newNode('D'));
    (root->child).push_back(newNode('E'));
    (root->child[0]->child).push_back(newNode('K'));
    (root->child[0]->child).push_back(newNode('J'));
    (root->child[2]->child).push_back(newNode('G'));
    (root->child[3]->child).push_back(newNode('C'));
    (root->child[3]->child).push_back(newNode('H'));
    (root->child[3]->child).push_back(newNode('I'));
    (root->child[0]->child[0]->child).push_back(newNode('N'));
    (root->child[0]->child[0]->child).push_back(newNode('M'));
    (root->child[3]->child[2]->child).push_back(newNode('L'));
 
    cout << diameter(root) << endl;
 
    return 0;
}

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Java

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// Java program to find the height of an N-ary
// tree
import java.util.*;
class GFG
{
 
// Structure of a node of an n-ary tree
static class Node
{
    char key;
    Vector<Node> child;
};
 
// Utility function to create a new tree node
static Node newNode(int key)
{
    Node temp = new Node();
    temp.key = (char) key;
    temp.child = new Vector<Node>();
    return temp;
}
 
// Utility function that will return the depth
// of the tree
static int depthOfTree(Node ptr)
{
    // Base case
    if (ptr == null)
        return 0;
 
    int maxdepth = 0;
 
    // Check for all children and find
    // the maximum depth
    for (Node it : ptr.child)
 
        maxdepth = Math.max(maxdepth,
                            depthOfTree(it));
 
    return maxdepth + 1;
}
 
// Function to calculate the diameter
// of the tree
static int diameter(Node ptr)
{
    // Base case
    if (ptr == null)
        return 0;
 
    // Find top two highest children
    int max1 = 0, max2 = 0;
    for (Node it : ptr.child)
    {
        int h = depthOfTree(it);
        if (h > max1)
        {
            max2 = max1;
            max1 = h;
        }
        else if (h > max2)
        max2 = h;
    }
 
    // Iterate over each child for diameter
    int maxChildDia = 0;
    for (Node it : ptr.child)
        maxChildDia = Math.max(maxChildDia,
                               diameter(it));
 
    return Math.max(maxChildDia, max1 + max2 + 1);
}
 
// Driver Code
public static void main(String[] args)
{
    /* Let us create below tree
    *         A
    *         / / \ \
    *     B F D E
    *     / \     | /|\
    *     K J G C H I
    *     /\         \
    * N M         L
    */
    Node root = newNode('A');
    (root.child).add(newNode('B'));
    (root.child).add(newNode('F'));
    (root.child).add(newNode('D'));
    (root.child).add(newNode('E'));
    (root.child.get(0).child).add(newNode('K'));
    (root.child.get(0).child).add(newNode('J'));
    (root.child.get(2).child).add(newNode('G'));
    (root.child.get(3).child).add(newNode('C'));
    (root.child.get(3).child).add(newNode('H'));
    (root.child.get(3).child).add(newNode('I'));
    (root.child.get(0).child.get(0).child).add(newNode('N'));
    (root.child.get(0).child.get(0).child).add(newNode('M'));
    (root.child.get(3).child.get(2).child).add(newNode('L'));
 
    System.out.print(diameter(root) + "\n");
}
}
 
// This code is contributed by Rajput-Ji

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C#

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// C# program to find the height of
// an N-ary tree
using System;
using System.Collections.Generic;
 
class GFG
{
 
// Structure of a node of an n-ary tree
class Node
{
    public char key;
    public List<Node> child;
};
 
// Utility function to create
// a new tree node
static Node newNode(int key)
{
    Node temp = new Node();
    temp.key = (char) key;
    temp.child = new List<Node>();
    return temp;
}
 
// Utility function that will return
// the depth of the tree
static int depthOfTree(Node ptr)
{
    // Base case
    if (ptr == null)
        return 0;
 
    int maxdepth = 0;
 
    // Check for all children and find
    // the maximum depth
    foreach (Node it in ptr.child)
 
        maxdepth = Math.Max(maxdepth,
                            depthOfTree(it));
 
    return maxdepth + 1;
}
 
// Function to calculate the diameter
// of the tree
static int diameter(Node ptr)
{
    // Base case
    if (ptr == null)
        return 0;
 
    // Find top two highest children
    int max1 = 0, max2 = 0;
    foreach (Node it in ptr.child)
    {
        int h = depthOfTree(it);
        if (h > max1)
        {
            max2 = max1;
            max1 = h;
        }
        else if (h > max2)
        max2 = h;
    }
 
    // Iterate over each child for diameter
    int maxChildDia = 0;
    foreach (Node it in ptr.child)
        maxChildDia = Math.Max(maxChildDia,
                               diameter(it));
 
    return Math.Max(maxChildDia,
                    max1 + max2 + 1);
}
 
// Driver Code
public static void Main(String[] args)
{
    /* Let us create below tree
    *         A
    *         / / \ \
    *     B F D E
    *     / \     | /|\
    *     K J G C H I
    *     /\         \
    * N M         L
    */
    Node root = newNode('A');
    (root.child).Add(newNode('B'));
    (root.child).Add(newNode('F'));
    (root.child).Add(newNode('D'));
    (root.child).Add(newNode('E'));
    (root.child[0].child).Add(newNode('K'));
    (root.child[0].child).Add(newNode('J'));
    (root.child[2].child).Add(newNode('G'));
    (root.child[3].child).Add(newNode('C'));
    (root.child[3].child).Add(newNode('H'));
    (root.child[3].child).Add(newNode('I'));
    (root.child[0].child[0].child).Add(newNode('N'));
    (root.child[0].child[0].child).Add(newNode('M'));
    (root.child[3].child[2].child).Add(newNode('L'));
 
    Console.Write(diameter(root) + "\n");
}
}
 
// This code is contributed by Rajput-Ji

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Output

7

Optimizations to above solution : 

We can make a hash table to store heights of all nodes. If we precompute these heights, we don’t need to call depthOfTree() for every node. 

A different optimized solution: Longest path in an undirected tree

Another Approach to get diameter using DFS in one traversal:

The diameter of a tree can be calculated as for every node

  • The current node isn’t part of diameter (i.e Diameter lies on of one of the children of the current node).
  • The current node is part of diameter (i.e Diameter passes through the current node).

Node: Adjacency List has been used to store the Tree.

Below is the implementation of the above approach:

C++

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// C++ implementation to find
// diameter of a tree using
// DFS in ONE TRAVERSAL
 
#include <bits/stdc++.h>
using namespace std;
#define maxN 10005
 
// The array to store the
// height of the nodes
int height[maxN];
 
// Adjacency List to store
// the tree
vector<int> tree[maxN];
 
// varaiable to store diameter
// of the tree
int diameter = 0;
 
// Function to add edge between
// node u to node v
void addEdge(int u, int v)
{
    // add edge from u to v
    tree[u].push_back(v);
 
    // add edge from v to u
    tree[v].push_back(u);
}
 
void dfs(int cur, int par)
{
    // Variables to store the height of children
    // of cur node with maximum heights
    int max1 = 0;
    int max2 = 0;
 
    // going in the adjacency list of the current node
    for (auto u : tree[cur]) {
         
        // if that node equals parent discard it
        if (u == par)
            continue;
 
        // calling dfs for child node
        dfs(u, cur);
 
        // calculating height of nodes
        height[cur] = max(height[cur], height[u]);
 
        // getting the height of children
        // of cur node with maximum height
        if (height[u] >= max1) {
            max2 = max1;
            max1 = height[u];
        }
        else if (height[u] > max2) {
            max2 = height[u];
        }
    }
 
    height[cur] += 1;
 
    // Diameter of a tree can be calculated as
    // diameter passing through the node
    // diameter doesn't includes the cur node
    diameter = max(diameter, height[cur]);
    diameter = max(diameter, max1 + max2 + 1);
}
 
// Driver Code
int main()
{
    // n is the number of nodes in tree
    int n = 7;
 
    // Adding edges to the tree
    addEdge(1, 2);
    addEdge(1, 3);
    addEdge(1, 4);
    addEdge(2, 5);
    addEdge(4, 6);
    addEdge(4, 7);
 
    // Calling the dfs function to
    // calculate the diameter of tree
    dfs(1, 0);
 
    cout << "Diameter of tree is : " << diameter - 1
         << "\n";
 
    return 0;
}

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Output

Diameter of tree is : 4

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Improved By : Rajput-Ji, pwnkumar0786

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