Trees with the same degree sequence and path numbers

We give an elementary procedure based on simple generating functions for constructing n (for any n >/2) pairwise non-isomorphic trees, all of which have the same degree sequence and the same number of paths of length k for all k >t 1. The construction can also be used to give a sufficient condition for isomorphism of caterpillars. In [2], a 2-variable polynomial that is closely related to the familiar Tutte polynomial of a graph or matroid is introduced and considered for trees. Two tree invariants are of special interest here. In particular, it is shown that for a given tree T, the polynomial f(T; t, z) determines the degree sequence of T as well as the number of paths of length k for all values of k t> 1. Thus, if f(T1) = f(T2), then the trees T1 and T2 must share the same degree sequence and the same number of paths of length k for all k (see Proposition 18 of [2]). In this context, it is natural to ask whether these two invariants uniquely determine the tree. We answer this question in the negative here, giving a procedure for constructing an infinite family of pairs of non-isomorphic trees, each pair of which has the same degree sequence and the same number of paths of length k for all k >~ 1. In fact, the construction can be used to create an arbitrarily large family of trees, all of which share the same degree sequence and same path numbers. This construction is completely elementary and also gives a sufficient condition for isomorphism of caterpillars. We assume the reader is familiar with graph theory; a standard reference is [1]. A caterpillar is a tree for which the set of vertices that are not leaves forms a path, called the spine. (See Fig. 1 for an example.) Let T be a caterpillar and let Uo,.