We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs
Galvez, M.; Tonks, A.; Kock, J. "Groupoids and Faà di Bruno Formulae for green functions in bialgebras of trees". 2012.