If you work with a single associative algebra $A$, then there is not so much sense to try to define a notion of $A$-tri-module. Namely, $A$-bimodules appear naturally as operadic $A$-modules.
To give you a picture, an associative algebra $A$ is in particular an algebra over the colored operad given by intervals of the real line and their disjoint union inclusions (a-k-a prefactorization algebras on the real line).
It has the property to be locally constant: the space associated to any interval is the same ($A$ itself).
Let me now relax a bit this locally constant property. Assume that you have such a (pre)factorization algebra over the real line such that it is locally constant outside the origin. Then you will get two associative algebras (one from $(0,+\infty)$ and the other from $(-\infty,0)$) together with a bunch of bimodules (One for each interval containing the origin).
There is nothing against the idea of having a factorization algebra on the graph consisting of three half-edges attached to a single vertex. Things like this might have appeared in $2d$ lattice models.
