Is it possible to come up with some kind of meaningful closed binary operation $\star$ on sets $C_n$ of Catalan objects? By Catalan objects I mean objects that correspond to a given Catalan number. I'll use well-nested parentheses since they're easy to write here, but feel free to choose your favourite objects.
When I say the operation should be meaningful, I mean that in the sense that it would be non-trivial and could be described in natural language just as Catalan objects can be described using a rule without simply listing them all. In other words, on the one hand, I don't want something trivial like every product gives the same result, and on the other hand, I don't want something so complex or unintuitive that you just have to look up the product on a seemingly arbitrary multiplication table.
One natural rule would be that $c \star c = c$ for all $c \in C$.
As an example, suppose we take the two possible ways to nest two sets of parentheses, $C_2 = \{()\ (),((\ ))\}$. We could decide that the level of nesting of a product must be the maximum level of the factors, so that $()\ () \star ((\ )) = ((\ )) \star ()\ () = ((\ ))$. Along with the rule that $c \star c = c$, this leads to a well-defined (and "meaningful") operation on $C_2$. But there is not enough information about how this operation works to generalize to $C_n$.
Any ideas?
Edit: Upon reflection, I see that there would be all sorts of natural operations, but the extent to which they are natural depends on the description of the objects. So maybe it would help if I said that I am particularly interested in binary trees. What now?