In this talk, I will present two problems related to the enumeration of trees. First, I will present a counting formula for multitype Cayley trees. This formula extends and unifies certain recent results of Bousquet-Melou and Chapuy. It has application to the multivariate Lagrange inversion formula and to the study of the profile of random trees. Second, I will present a combinatorial proof of a counting formula for the spanning trees of the hypercube. It highlights an unexpected independence property for the directions of edges in the uniform spanning tree of the hypercube.

Our proofs for both problems takes advantage of certain symmetries of the enumerative formulas: we first prove these symmetries by simple combinatorial arguments, and then deduce the general formulas from particular cases.

The first part is joint work with Alejandro Morales.