LeetCode Q310: Minimum Height Tree

For a undirected graph with tree characteristics, we can choose any node as the root. The result graph is then a rooted tree. Among all possible rooted trees, those with minimum height are called minimum height trees (MHTs). Given such a graph, write a function to find all the MHTs and return a list of their root labels.

Format
The graph contains n nodes which are labeled from 0 to n - 1. You will be given the number n and a list of undirected edges (each edge is a pair of labels).

You can assume that no duplicate edges will appear in edges. Since all edges are undirected, [0, 1] is the same as [1, 0] and thus will not appear together in edges.

Example 1:

Given n = 4, edges = [[1, 0], [1, 2], [1, 3]]

        0
        |
        1
       / \
      2   3

return [1]

Example 2:

Given n = 6, edges = [[0, 3], [1, 3], [2, 3], [4, 3], [5, 4]]

     0  1  2
      \ | /
        3
        |
        4
        |
        5

return [3, 4]

Solution:

The first question we need to ask ourself is How many MHTs can a graph have at most?

To answer this question, we should go back to the definition of MHT again, and think carefully, what MHT exactly is and which nodes are most likely the roots of MHTs. 

As how MHT is defined, MHTs are trees whose longest route are shortest among all converted trees. That means, in a MHT, we shouldn't have some root-to-leave routes which are much much longer than other routes, otherwise, we could shift out root a few steps to further lower tree's height. Actually, it is best to have all routes having the same length! This is a very important observation you should keep in mind. Then, how do we locate those nodes which have root-to-leave routes with almost the same length? Th easiest way is to setup pointers at each leave in current iteration, and remove these leaves every iteration, until there are most TWO nodes left. The nodes left are roots of MHTs we needed.

So, we will have at most two MHTs in a any given tree graph.