  We study the pre-Lie algebra of rooted trees $(\Cal T, \rightarrow)$ and we
define a pre-Lie structure on its doubling space $(V, \leadsto)$. Also, we find
the enveloping algebras of the two pre-Lie algebras denoted respectively by
$(\Cal H', \star, \Gamma)$ and $(\Cal D', \bigstar, \chi)$. We prove that
$(\Cal D', \bigstar, \chi)$ is a module-bialgebra on $(\Cal H', \star, \Gamma)$
and we find some relations between the two pre-Lie structures.
