Deleting to Structured Trees

Feedback Vertex Set asks for the fewest vertex deletions that leave a forest. I ask for more structure in what remains: delete a few vertices (or a few edges) so that the graph becomes a full binary tree, the kind found in phylogenetic trees and in some polymerization processes. Both versions are NP-hard. The edge version is the surprise. Deleting edges of minimum total weight to leave some tree is just a spanning tree computation, yet asking for a full binary tree makes the problem hard. The proofs build on a warm-up reduction for ternary trees. On the algorithmic side, both problems are FPT in the number of deletions.