Résumé
Advances in Applied Mathematics 47 (2011) 282-308 Hopf algebra structures on rooted trees are by now a well-studied object,
especially in the context of combinatorics. In this work we consider a Hopf
algebra H by introducing a coproduct on a (commutative) algebra of rooted
forests, considering each tree of the forest (which must contain at least one
edge) as a Feynman-like graph without loops. The primitive part of the graded
dual is endowed with a pre-Lie product defined in terms of insertion of a tree
inside another. We establish a surprising link between the Hopf algebra H
obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted
trees by means of a natural H-bicomodule structure on the latter. This enables
us to recover recent results in the field of numerical methods for differential
equations due to Chartier, Hairer and Vilmart as well as Murua.