  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.
