Choose minimum subset of edges in tree that connects all important nodes
Note this is equivalently finding edges with minimum sum of weights such that their removal leaves each important node in a separate component, which is called minimum multiway cut problem. 

This problem is NP-hard in general graph, while Chopra, Sunil, and Mendu R. Rao developed a polynomial time algorithm for tree in 1991 [1].



[1] Chopra, Sunil, and Mendu R. Rao. "On the multiway cut polyhedron." Networks 21.1 (1991): 51-89.