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.

中文翻译：

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在具有相同受限 U 多项式和有根 U 多项式的最小树上

在本文中，我们构造了具有相同受限 $U$ 多项式的非同构树对的显式示例，其中每个 $k$；我们的意思是多项式的度数最多为 $k+1$。这种构造的主要工具是将 $U$ 多项式推广到有根图，我们将在本文中介绍和研究。最值得注意的是，我们证明了有根树可以从它的有根 $U$ 多项式重建。