INTERCHANGING BRANCHES AND SIMILARITY IN A TREE

A shoot is a fixed subset of branches rooted at a given vertex of a tree. We show that interchanging two nonintersecting shoots is an isomorphism of a tree only in two trivial cases: when either the shoots are isomorphic as rooted trees or their roots are similar in a tree obtained by deleting the shoots without the roots. The proof is based on a sufficient condition for similarity of two vertices in a tree. We also consider some applications of the above results to problems concerning Number Deck reconstruction of a tree.