Groupoids and Faa di Bruno formulae for Green functions in bialgebras of   trees
  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. 
Imma Galvez-Carrillo, Joachim Kock, Andrew Tonks
math.QA math-ph math.CT math.MP