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:
Example 2:
Prerequisite: Diameter of a binary tree.
The path can either start from one of the nodes and go 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++
// 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;
}
Java
// 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
Python3
# Python program to find the height of an N-ary
# tree
# Structure of a node of an n-ary tree
class Node:
def __init__(self, x):
self.key = x
self.child = []
# Utility function that will return the depth
# of the tree
def depthOfTree(ptr):
# Base case
if (not ptr):
return 0
maxdepth = 0
# Check for all children and find
# the maximum depth
for it in ptr.child:
maxdepth = max(maxdepth , depthOfTree(it))
return maxdepth + 1
# Function to calculate the diameter
# of the tree
def diameter(ptr):
# Base case
if (not ptr):
return 0
# Find top two highest children
max1, max2 = 0, 0
for it in ptr.child:
h = depthOfTree(it)
if (h > max1):
max2, max1 = max1, h
elif (h > max2):
max2 = h
# Iterate over each child for diameter
maxChildDia = 0
for it in ptr.child:
maxChildDia = max(maxChildDia, diameter(it))
return max(maxChildDia, max1 + max2 + 1)
# Driver program
if __name__ == '__main__':
# /* Let us create below tree
# * A
# * / / \ \
# * B F D E
# * / \ | /|\
# * K J G C H I
# * /\ \
# * N M L
# */
root = Node('A')
(root.child).append(Node('B'))
(root.child).append(Node('F'))
(root.child).append(Node('D'))
(root.child).append(Node('E'))
(root.child[0].child).append(Node('K'))
(root.child[0].child).append(Node('J'))
(root.child[2].child).append(Node('G'))
(root.child[3].child).append(Node('C'))
(root.child[3].child).append(Node('H'))
(root.child[3].child).append(Node('I'))
(root.child[0].child[0].child).append(Node('N'))
(root.child[0].child[0].child).append(Node('M'))
(root.child[3].child[2].child).append(Node('L'))
print(diameter(root))
# This code is contributed by mohit kumar 29C#
// 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
Javascript
<script>
// Javascript program to find the
// height of an N-ary tree
// Structure of a node of an n-ary tree
class Node{
// Utility function to create a new tree node
constructor(key)
{
this.key=key;
this.child=[];
}
}
// Utility function that will
// return the depth
// of the tree
function depthOfTree(ptr)
{
// Base case
if (ptr == null)
return 0;
let maxdepth = 0;
// Check for all children and find
// the maximum depth
for (let it=0;it< ptr.child.length;it++)
maxdepth = Math.max(maxdepth,
depthOfTree(ptr.child[it]));
return maxdepth + 1;
}
// Function to calculate the diameter
// of the tree
function diameter(ptr)
{
// Base case
if (ptr == null)
return 0;
// Find top two highest children
let max1 = 0, max2 = 0;
for (let it=0;it< ptr.child.length;it++)
{
let h = depthOfTree(ptr.child[it]);
if (h > max1)
{
max2 = max1;
max1 = h;
}
else if (h > max2)
max2 = h;
}
// Iterate over each child for diameter
let maxChildDia = 0;
for (let it=0;it< ptr.child.length;it++)
maxChildDia = Math.max(maxChildDia,
diameter(ptr.child[it]));
return Math.max(maxChildDia, max1 + max2 + 1);
}
// Driver Code
/* Let us create below tree
* A
* / / \ \
* B F D E
* / \ | /|\
* K J G C H I
* /\ \
* N M L
*/
let root = new Node('A');
(root.child).push(new Node('B'));
(root.child).push(new Node('F'));
(root.child).push(new Node('D'));
(root.child).push(new Node('E'));
(root.child[0].child).push(new Node('K'));
(root.child[0].child).push(new Node('J'));
(root.child[2].child).push(new Node('G'));
(root.child[3].child).push(new Node('C'));
(root.child[3].child).push(new Node('H'));
(root.child[3].child).push(new Node('I'));
(root.child[0].child[0].child).push(new Node('N'));
(root.child[0].child[0].child).push(new Node('M'));
(root.child[3].child[2].child).push(new Node('L'));
document.write(diameter(root) + "\n");
// This code is contributed by patel2127
</script>
Output
7
Optimizations to above solution :
We can find diameter without calculating depth of the tree making small changes in the above solution, similar to finding diameter of binary tree.
C++
// 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;
}
int diameter(struct Node *ptr,int &diameter_of_tree)
{
// 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 = diameter(*it,diameter_of_tree);
if (h > max1)
max2 = max1, max1 = h;
else if (h > max2)
max2 = h;
}
// Find whether our node can be part of diameter
diameter_of_tree = max(max1 + max2 + 1,diameter_of_tree);
return max(max1,max2) + 1;
}
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'));
// for storing diameter
int diameter_of_tree = 0;
diameter(root,diameter_of_tree);
cout << diameter_of_tree << endl;
return 0;
}
// This code is improved by bhuvanJava
// 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;
}
// for storing diameter_of_tree
public static int diameter_of_tree = 0;
// 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 = diameter(it);
if (h > max1) {
max2 = max1;
max1 = h;
}
else if (h > max2)
max2 = h;
}
diameter_of_tree
= Math.max(max1 + max2 + 1, diameter_of_tree);
return (Math.max(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'));
diameter(root);
System.out.print(diameter_of_tree + "\n");
}
}
// This code is improved by BhuvanOutput
7
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 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++
// 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];
// variable 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;
}Java
/*package whatever //do not write package name here */
import java.io.*;
import java.util.*;
class GFG {
static int maxN = 10005;
// The array to store the
// height of the nodes
static int[] height = new int[maxN];
// Adjacency List to store
// the tree
static ArrayList<ArrayList<Integer>> tree = new ArrayList<ArrayList<Integer>>();
// variable to store diameter
// of the tree
static int diameter = 0;
// Function to add edge between
// node u to node v
static void addEdge(int u, int v)
{
// add edge from u to v
tree.get(u).add(v);
// add edge from v to u
tree.get(v).add(u);
}
static 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 (int u : tree.get(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] = Math.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 = Math.max(diameter, height[cur]);
diameter = Math.max(diameter, max1 + max2 + 1);
}
public static void main (String[] args)
{
for(int i = 0; i < maxN; i++)
{
tree.add(new ArrayList<Integer>());
}
// 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);
System.out.println("Diameter of tree is : " +(diameter - 1));
}
}
// This code is contributed by ab2127.Python3
# C++ implementation to find
# diameter of a tree using
# DFS in ONE TRAVERSAL
maxN = 10005
# The array to store the
# height of the nodes
height = [0 for i in range(maxN)]
# Adjacency List to store
# the tree
tree = [[] for i in range(maxN)]
# variable to store diameter
# of the tree
diameter = 0
# Function to add edge between
# node u to node v
def addEdge(u, v):
# add edge from u to v
tree[u].append(v)
# add edge from v to u
tree[v].append(u)
def dfs(cur, par):
global diameter
# Variables to store the height of children
# of cur node with maximum heights
max1 = 0
max2 = 0
# going in the adjacency list of the current node
for u in 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]
elif(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
# n is the number of nodes in tree
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)
print("Diameter of tree is :", diameter - 1)
# This code is contributed by avanitrachhadiya2155Javascript
<script>
// Javascript implementation to find
// diameter of a tree using
// DFS in ONE TRAVERSAL
let maxN = 10005;
// The array to store the
// height of the nodes
let height=new Array(maxN);
// Adjacency List to store
// the tree
let tree=new Array(maxN);
for(let i=0;i<maxN;i++)
{
height[i]=0;
tree[i]=[];
}
// variable to store diameter
// of the tree
let diameter = 0;
// Function to add edge between
// node u to node v
function addEdge(u,v)
{
// add edge from u to v
tree[u].push(v);
// add edge from v to u
tree[v].push(u);
}
function dfs(cur,par)
{
// Variables to store the height of children
// of cur node with maximum heights
let max1 = 0;
let max2 = 0;
// going in the adjacency list
// of the current node
for (let u=0;u<tree[cur].length;u++) {
// if that node equals parent discard it
if (tree[cur][u] == par)
continue;
// calling dfs for child node
dfs(tree[cur][u], cur);
// calculating height of nodes
height[cur] = Math.max(height[cur],
height[tree[cur][u]]);
// getting the height of children
// of cur node with maximum height
if (height[tree[cur][u]] >= max1) {
max2 = max1;
max1 = height[tree[cur][u]];
}
else if (height[tree[cur][u]] > max2) {
max2 = height[tree[cur][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 = Math.max(diameter, height[cur]);
diameter = Math.max(diameter, max1 + max2 + 1);
}
// Driver Code
// n is the number of nodes in tree
let 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);
document.write("Diameter of tree is : " +(diameter - 1))
// This code is contributed by unknown2108
</script>
Output
Diameter of tree is : 4
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