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?

For $k = 0$ to $n - 1$ (or vice versa) enumerate all the possible left subtrees with $k$ nodes. For each possible left subtree with $k$ nodes, enumerate all the possible right subtrees with $n - k - 1$ nodes. Recursively, this gives for every positive integer $n$ a fairly natural way to totally order the set of all unlabeled binary trees with $n$ nodes, by referring to the same procedure to totally order the set of all trees with any given smaller number of nodes.
