Minimize the Diameter after Joining Two Trees
Company: Tekion
Role: Software Engineer
Category: Coding & Algorithms
Difficulty: hard
Interview Round: Other
### Problem
You are given two undirected trees. The first has `n` nodes labeled `0` through `n - 1`, and the second has `m` nodes labeled `0` through `m - 1`.
The rows of `edges1` and `edges2` list the edges of their respective trees. You must add exactly one edge joining a node of the first tree to a node of the second tree.
Return the smallest possible diameter of the resulting tree. A tree's diameter is the greatest distance between any two nodes, measured in edges.
```cpp
class Solution {
public:
int minimumDiameterAfterMerge(vector<vector<int>>& edges1, vector<vector<int>>& edges2);
};
```
### Examples
**Example 1:**
```text
Input: edges1 = [[0,1],[1,2],[2,3]], edges2 = [[0,1],[1,2]]
Output: 4
Explanation: Joining node 1 of the first tree to node 1 of the second gives a diameter of 4. No choice of the joining edge gives a smaller diameter.
```
**Example 2:**
```text
Input: edges1 = [], edges2 = []
Output: 1
Explanation: Each input is a separate one-node tree. The required joining edge makes their distance 1.
```
### Constraints
- `1 <= n, m <= 100000`.
- `edges1.length == n - 1` and `edges2.length == m - 1`.
- Each row in either edge list contains exactly two integers.
- For every `[a_i, b_i]` in `edges1`, `0 <= a_i, b_i < n`.
- For every `[u_i, v_i]` in `edges2`, `0 <= u_i, v_i < m`.
- Both edge lists represent valid trees.
### Clarifying Questions
- Does label `0` refer to the same node in both inputs? No. The two trees have separate node sets.
- Can an input edge list be empty? Yes, when that tree has one node.
- Is the answer a number of nodes or edges? Edges.
- May an original edge be removed or more than one edge added? No. Keep all original edges and add exactly one joining edge.
- Must the endpoints of that edge be returned? No, return only the minimum diameter.
### What a Strong Answer Covers
- Considers distances both within an original tree and between the two trees.
- Justifies that no different joining edge can produce a smaller answer.
- Handles one-node trees and the full combined input size.
```hint Separate the possible paths
A path either stays inside one original tree or crosses the new edge. How does each chosen endpoint affect the longest path of the second kind?
```
Overview: Join two undirected trees with exactly one edge and return the minimum possible diameter. Distances count edges, and either input may be a single-node tree.