My question is a special case of a question asked here, but I think/hope that this particular version admits an elementary solution.
For each integer $h \geq0$, there is a unique full, complete, binary tree of height $h$. Those are a lot of adjectives, but the tree they describe is probably the first one you'd draw when you heard the term "binary tree".
I'm interested in the set $\mathcal S_h$ of full subtrees of that tree. Recall that a binary tree is called $\ {full}$ if each of its non-leaf nodes have exactly two children.
I'd like to know:
(1) the size of $\mathcal S_h$
and, more importantly,
(2) an enumeration of $\mathcal S_h$
Either the full subtree is just the root, or it is the root, both branches from the root, and the two subtrees below, which must also be full subtrees. Thus, if $S_n$ is the number you are after, we have $S_n=S_{n-1}^2+1$, with $S_0=1$. ($S_1=2, S_2=5, S_3=26$ etc.) It seems to be this sequence: A003095, offset by one.
As for enumeration, I guess you can enumerate the root first, and then in a double loop enumerate pairs of subtrees on the left and on the right.