Is there a natural way to totally order the set of unlabeled binary trees on $n$ nodes? Let $C_n$ be the $n^{th}$ Catalan number. There are $C_n$ unlabeled binary trees having $n$ internal nodes.  I want to totally order these trees in some (hopefully not too complicated) natural manner.  Perhaps there is some "standard" way of doing this.  When I look at pictures of, say the 14 binary trees on 4 nodes, they never seem to be listed in any particular order.  Notice that what I want to order is the trees themselves not the vertices within each tree.
I tried classifying by height and then by height of shortest branches from left to right but I'm not sure if what I have is correct.  I think there must be some well known way to order these trees?

Let $S$ and $T$ be binary trees. Provided $S$ and $T$ are nontrivial, we write $S=(S^L,S^R)$ and $T=(T^L,T^R)$. An ordering on trees is determined recursively as follows:
$$
S\le T\iff \begin{cases}
S=\text{trivial tree}&\text{or}\\
S^L<T^L & \text{or}\\
S^L=T^L\text{ and }S^R\le T^R
\end{cases}
$$
Roughly, trees are ordered in terms of simplicity, with preference given to the left subtree, so this is a recursive lexicographic order. For example, when $n=3$,

