On The Smallest Trees With The Same Restricted U-Polynomial And The Rooted U-Polynomial

In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted U-polynomial for every k; by this we mean that the polynomials agree on terms with degree at most k+1. The main tool for this construction is a generalization of the U-polynomial to rooted graphs, which we introduce and study in this article. Most notably we show that rooted trees can be reconstructed from its rooted U-polynomial. (C) 2020 Elsevier B.V. All rights reserved.