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See J. W. Moon's Counting Labelled Trees, equation (2.2), math.ucla.edu/~pak/hidden/papers/…. This book is a great resource for anything to do with enumeration of labeled trees.
The paper "Cyclotomic factors of the descent set polynomial" by Denis Chebikin, Richard Ehrenborg, Pavlo Pylyavskyy, and Margaret Readdy, J. Combin. Theory Ser. A. 116 (2009), 247–264, sciencedirect.com/science/article/pii/S0097316508000897 may be relevant. They consider the number of odd values of $\beta_n(S)$.
Graphs (undirected) with the property that no two distinct vertices have the same neighborhood have several names: point-determining, R-thin, mating type, and M-graphs. See, e.g., doi.org/10.1016/j.jcta.2010.03.009. I don't know that the analogous digraphs have been studied.
