Minimum path to connect k nodes in a graph with n nodes

Question

Given a graph with n nodes and n-1 edges, find the lowest amount of edges needed to connect k nodes. You are given m numbers k <= n for which you have to solve the problem. The edges are unweighted and the graph does not necessarily describe a binary tree.

For a graph with n=5 and k=4, a possible path could be 1-2-5-2-3.

My approach was a greedy method in which I start from the node with the highest rank and try to add edges from there, but I did not have much success in trying to formulate an algorithm.

Answer 1

Try to find longest path in your tree, you will take every edge on that route once, than choose random node that has an edge, that connect that node to already choosen elements.

So find longest path in your tree if that path has k or more nodes it is the end, you choose element on that path, every edge will be taken once
Else choose a random, untaken already node, that is connected by any edge directly to one of already taken nodes, as long as you do not have k nodes in your collection.