  We extend the results we obtained in an earlier work. The cocommutative case
of rooted ladder trees is generalized to a full Hopf algebra of (decorated)
rooted trees. For Hopf algebra characters with target space of Rota-Baxter
type, the Birkhoff decomposition of renormalization theory is derived by using
the Rota-Baxter double construction, respectively Atkinson's theorem. We also
outline the extension to the Hopf algebra of Feynman graphs via decorated
rooted trees.
