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Equivalence classes of nodes in trees and rational generating functions

Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied to counting the number of conjugacy classes of commuting tuples in finite groups and the number of isomorphism classes of representations of polynomial algebras over finite fields. The method for computing the rational generating functions, when applied to the study of point configurations in finite sets, gives rise to some classical combinatorial results on Bell numbers and Stirling numbers of the second kind. When applied to the study of vector configurations in a finite vector space, it reveals a connection between counting such configurations and Gaussian binomial coefficients.