  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 of\H. 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.

Rings and Algebras