On arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertexdisjoint subtrees of T whose orders are given by τ ; from a result by Barth and Fournier it follows that ∆(T ) ≤ 4. A necessary and a sufficient condition for being an arbitrarily vertex decomposable star-like tree have been exhibited. The conditions seem to be very close to each other.