Iterate through every possible subtree by doing a bitmask on which vertices to include. How can you determine if a subtree is valid (all vertices are connected)?

To determine connectivity, count the number of reachable vertices starting from any included vertex and only traveling on edges connecting 2 vertices in the subtree. The count should be the same as the number of 1s in the bitmask.

The diameter is basically the maximum distance between any two nodes. Root the tree at a vertex. The answer is the max of the heights of the two largest subtrees or the longest diameter in any of the subtrees.