Convex extendable trees (Q1304821)

A graph is distance convex simple if all its nontrivial convex vertex sets (i.e. containing all shortest paths between any of its pairs) are pairs. A tree is convex extendable if it is a spanning tree of a convex simple graph. It is shown that all trees up to order nine are convex extendable, as well as any tree of diameter three or five, and those of diameter four of which the central vertex has even degree. A similar question for minimal path convexity (replacing shortest paths by chordless paths) is also investigated.