Abstract: A commutative Hopf algebra on forests of rooted trees was introduced by D. Kreimer in order to describe the process of renormalization in quantum field theory. A noncommutative version of this Hopf algebra was described by L. Foissy. We shall discuss how these Hopf algebras are related, and how the solution of a combinatorial Dyson-Schwinger equation in Kreimer's Hopf algebra can be simplified by lifting it to Foissy's Hopf algebra.