Inversion of integral series enumerating planar trees
  We consider an integral series f(X,t) which depends on the choice of a set X of labelled planar rooted trees. We prove that its inverse for composition is of the form f(Z,t) for another set Z of trees, deduced from X. The proof is self-contained, though inspired by the Koszul duality theory of quadratic operads. 
Jean-Louis Loday