Abstract
We study the pre-Lie algebra of rooted trees [Formula: see text] and we define a pre-Lie structure on its doubling space [Formula: see text]. Also, we find the enveloping algebras of the two pre-Lie algebras denoted, respectively, by [Formula: see text] and [Formula: see text]. We prove that [Formula: see text] is a module-bialgebra on [Formula: see text] and we find some relations between the two pre-Lie structures.