  This paper focuses on the well-known problem due to Stanley of whether two
non-isomorphic trees can have the same $U$-polynomial (or, equivalently, the
same chromatic symmetric function). We consider the $U_k$-polynomial, which is
a restricted version of $U$-polynomial, and construct with the help of
solutions of the Prouhet-Tarry-Escott problem, non-isomorphic trees with the
same $U_k$-polynomial for any given $k$. By doing so, we also find a new class
of trees that are distinguished by the $U$-polynomial up to isomorphism.
