On arbitrarily vertex decomposable trees

A tree TT is arbitrarily vertex decomposable if for any sequence ττ of positive integers adding up to the order of TT there is a sequence of vertex-disjoint subtrees of TT whose orders are given by ττ; from a result by Barth and Fournier it follows that Δ(T)⩽4Δ(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.