Hi -

Some comments:

1) In your initial definition you don't say the identity morphism is called $\mathrm{Id}_A : A \to A$. This is presumably a deliberate trick to make things easier to read. If so, fine.

2) You also don't include a notation for composition, presumably for the same reason. If so, fine.

3) You don't actually draw the "tree diagram" as a tree. I would find it impossible to truly understand operads without seeing such trees (and drawing them). Here is my attempt to explain operads - it's packed with such pictures:

* Review of _[Operads in Algebra, Topology and Physics](http://math.ucr.edu/home/baez/operad.pdf)_.

By the way, I was only talking about operads with one object, so you don't see objects labelling the "wires" or "edges" of these pictures. Also, I accidentally left out one of the laws governing operads.