  We first study tensor products of tridendriform algebras in order to
introduce the notion of tridendriform bialgebra. We shall need for this a
notion of augmented tridendriform algebras. Inspired by the work of J-L. Loday
and M. Ronco, we build free tridendriform algebras over reduced trees and show
that they have a coproduct satisfying some compatibilities with the
tridendriform products. Such an object will be called a (3, 1)--dendriform
algebra. Studying the free (3, 1)--dendriform bialgebra over one generator, we
describe its products and coproduct in a combinatorial way. The products are
described by branches shuffle and the coproduct by admissible cuts. We compare
it with quasi-shuffle algebras over words. Its graded dual is the bialgebra
TSym introduced by N. Bergeron and al which is described by the lightening
splitting of a tree. As a consequence, this shows that TSym has a (1,
3)--dendriform bialgebra structure. This means that its coproduct can be split
in three parts with convenient compatibilities. This can be extended to (3,
1)-bialgebras over an arbitrary number of generators. Finally, we introduce the
notion of (3, 2)--dendriform bialgebra. This is a Hopf algebra, where we can
split the product in three pieces and the coproduct in two with Hopf
compatibilities. We give an example of such an algebra built on the free (3,
1)-dendriform bialgebra with one generator. We describe and generate its
codendriform primitives and count its coassociative primitives thanks to L.
Foissy's work. We end this paper by showing that a quotient of this (3,
2)-dendriform bialgebra is the Loday-Ronco bialgebra.

Combinatorics