  We here both unify and generalize nonassociative structures on typed binary
trees, that is to say plane binary trees which edges are decorated by elements
of a set $\Omega$. We prove that we obtain such a structure, called an
$\Omega$-dendriform structure, if $\Omega$ has four products satisfying certain
axioms (EDS axioms), including the axioms of a diassociative semigroup. This
includes matching dendriform algebras introduced by Zhang, Gao and Guo and
family dendriform algebras associated to a semigroup introduced by Zhang, Gao
and Manchon , and of course dendriform algebras when $\Omega$ is reduced to a
single element. We also give examples of EDS, including all the EDS of
cardinality two; a combinatorial description of the products of such a
structure on typed binary trees, but also on words; a study of the Koszul dual
of the associated operads; and considerations on the existence of a coproduct,
in order to obtain dendriform bialgebras.
