  We describe the proalgebraic groups represented by three Hopf algebras on
planar binary trees previously introduced by the author and Christian Brouder
in relation with the renormalization of quantum electrodynamics. Using two
monoidal structures and a set-operad structure on planar binary trees, we show
that these groups can be realized on formal series expanded over trees, and
that the group laws are generalization of the multiplication and the
composition of usual series in one variable. All the constructions are done in
a general operad-theoretic setting, and then applied to the duplicial operad on
trees.
