Diameter of N-Ary Tree

class Solution {
    protected int maxDiameter = 0;

    protected int depthCalc(Node node, int depth) {
        if (node.children.size() == 0)
            return depth;

        int deepest = depth, secondDeepest = 0;
        for (Node child : node.children) {
            int childDepth = depthCalc(child, depth + 1);
            if (childDepth > deepest) {
                secondDeepest = deepest;
                deepest = childDepth;
            } else if (childDepth > secondDeepest) {
                secondDeepest = childDepth;
            }
            int currentDistance = deepest + secondDeepest - 2 * depth;
            this.maxDiameter = Math.max(this.maxDiameter, currentDistance);
        }

        return deepest;
    }

    public int getDiameter(Node root) {
        this.maxDiameter = 0;
        depthCalc(root, 0);
        return maxDiameter;
    }
}

The provided Java solution calculates the diameter of an N-Ary tree, where the diameter is defined as the longest path between any two nodes in the tree. The code defines a Solution class that includes methods for determining this diameter.

The primary logic involves two methods in the class:

• depthCalc(Node node, int depth) computes the depth of the tree recursively and tracks the two deepest children at each node to calculate the potential diameter at that node. It updates the maxDiameter attribute with the largest diameter found during traversal.
• getDiameter(Node root) initializes the maximum diameter to zero and starts the depth calculation from the root.

In depthCalc, a check is made for nodes without children (leaf nodes), returning their current depth as the base case for recursion. For non-leaf nodes, the method iterates over all children to identify the deepest and the second deepest paths. These values are used to update the maxDiameter.

Steps to calculate the diameter:

1. Initialize deepest and secondDeepest for storing the maximum depths of the first two children respectively.
2. Traverse through all children of the current node to calculate their depths by recursive calls.
3. Determine if the current child's depth surpasses the depths stored in deepest or secondDeepest, updating those variables as necessary.
4. Calculate the potential diameter as the sum of the two deepest paths minus double the current depth to account for the actual paths between these nodes.
5. Compare and update maxDiameter with the newly computed diameter if it's larger.
6. Return the deepest value to the parent call, providing depth information for parent nodes to compute their diameters.

This approach ensures that the entire tree is traversed to find the longest possible path between pair of nodes, thereby determining the tree's diameter in an efficient manner using depth-first search principles.

"""
# Definition for a TreeNode with potential multiple children.
class TreeNode:
    def __init__(self, value=None, children_list=None):
        self.value = value
        self.children = children_list if children_list is not None else []
"""
class Solution:
    def maxDiameter(self, root: 'TreeNode') -> int:
        """
        :type root: 'TreeNode'
        :rtype: int
        """
        diameter = 0

        def maxDepthFromNode(node, depth):
            """ Calculate max depth starting from this node all the way to the leaf nodes """
            nonlocal diameter

            if len(node.children) == 0:
                return depth
            
            max_1st_depth, max_2nd_depth = depth, 0
            for child in node.children:
                child_depth = maxDepthFromNode(child, depth + 1)
                if child_depth > max_1st_depth:
                    max_1st_depth, max_2nd_depth = child_depth, max_1st_depth
                elif child_depth > max_2nd_depth:
                    max_2nd_depth = child_depth

            # compute the distance between the two most distant leaf nodes
            path_length = max_1st_depth + max_2nd_depth - 2 * depth
            diameter = max(diameter, path_length)

            return max_1st_depth

        maxDepthFromNode(root, 0)
        return diameter

This Python solution calculates the diameter of an N-ary tree. The diameter of a tree is the length of the longest path between any two nodes in that tree.

• TreeNode Class: A custom class to structure each node of the N-ary tree. Each node has a value and a list of children nodes.
• maxDiameter Method: Computes the maximum diameter of the tree starting from the root.
• maxDepthFromNode Helper Function: This recursive function calculates the maximum depths diverging from each node to determine the longest paths possible, updating the diameter at each step. It uses two variables, max_1st_depth and max_2nd_depth, to hold the maximum two lengths found among the children to calculate potential maximum path lengths effectively.
• Computing Path Length and Updating Diameter: Whenever two maximum depths are found (from any node to its descendent leaves), the potential diameter through that node (root of the subpaths) is determined.

The main function, maxDiameter, initializes the longest diameter variable and starts the recursive depth calculation from the root. It eventually returns the maximum diameter found during the traversal. This approach ensures a comprehensive evaluation through all potential paths in the N-ary tree, comparing path lengths to determine the maximum effectively.