  We introduce a non-symmetric operad $\mathcal{N}$, whose dimension in degree
$n$ is given by the Catalan number $c_{n-1}$. It arises naturally in the study
of coalgebra structures defined on compatible associative algebras. We prove
that any free compatible associative algebra admits a compatible infinitesimal
bialgebra structure, whose subspace of primitive elements is a
$\mathcal{N}$-algebra. The data $({\rm As},{\rm As}^2, \mathcal{N})$ is a good
triple of operads, in J.-L. Loday's sense. Our construction induces another
triple of operads $({\rm As},{\rm As}_2,{\rm As})$, where ${\rm As}_2$ is the
operad of matching dialgebras. Motivated by A. Goncharov's Hopf algebra of
paths $P(S)$, we introduce the notion of bi-matching dialgebras and show that
the Hopf algebra $P(S)$ is a bi-matching dialgebras.
