DP on Trees | Set-3 ( Diameter of N-ary Tree )

Given an N-ary tree, the task is to calculate the longest path between any two nodes (also known as the diameter of the tree).

Examples:

Example1: The diameter of the N-ary tree in the below example is 9, which represents the number of edges along the path marked with green nodes.

👁 diameter---------of---------a---------n---------array---------tree

Example2: The diameter of the N-ary tree in the below example is 7, which represents the number of edges along the path marked with green nodes.

👁 diameter---------of---------a---------n---------array---------tree---------2

Approach:

The idea is to use Dynamic Programming with two key states: dp1[node] and dp2[node]. The value dp1[node] represents the longest path from the current node to any leaf in its subtree, while dp2[node] represents the longest path passing through the current node and two of its subtrees. By combining these two states across all nodes, we compute the maximum diameter. A Depth First Search (DFS) traversal is used to calculate these states efficiently.

Step-by-step approach:

• Use dp1[node] (longest path to a leaf in subtree) and dp2[node] (longest path passing through the node and its two subtrees).
• For leaf nodes, set dp1[node] = 0 and dp2[node] = 0.
• Begin a DFS traversal from an arbitrary node, tracking the parent to avoid revisits.
• For each node, find the two largest dp1[child] values using firstMaxand secondMax.
• Compute dp1[node] = 1 + firstMax and dp2[node] = firstMax + secondMax.
• Update the diameter as the maximum of dp2[node] and the current diameter.

3

Time Complexity: O(n), where n is the total number of nodes in the tree.
Auxiliary Space: O(n)

Related articles:

• Diameter of an N-ary tree
• Diameter of n-ary tree using BFS