Groups of tree-expanded series

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.