Given a directed Cartesian product I of locally finite, leafless, rooted directed trees I1, . . ., Id of finite joint branching index, one may associate with I the Drury-Arveson-type C[z1, . . ., zd]-Hilbert module ℋca (I) of vector-valued holomorphic functions on the unit ball Bd in Cd, where a > 0. The main result of this paper classifies all directed Cartesian products I for which the Hilbert modules ℋca (I ) are isomorphic in case a is an integer. Indeed, a careful analysis of these Hilbert modules allows us to prove that the cardinality of generations of I1, . . ., Id are complete invariants for ℋca (·) if ad ≠ 1. © THETA, 2019.