Fa`a di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B^+(P(X)) in the non-commutative Connes-KreimerHopf algebra of planar rooted trees H, where B^+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A_P ofH. We describe all the formal series P such that A_P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H, organized into three isomorphism classes: a first one, restricted to a olynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Fa`a di Bruno Hopf algebra. By taking the quotient, the last classe gives an infinite set of embeddings of the Fa`a di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Fa`a di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, togetherwith a non commutative version of this embedding.