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I prefer to enumerate via a shortlex variant, primary key being number of leaves and secondary being the index of the particular expansion.

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Why the preference?

(For context, I'm completely new to this topic.)

Because I use them for testing, so I generally want to uniformly generate samples within a particular size of tree.

(generating random trees the naive way tends to produce the occasional huge tree amidst huge numbers of tiny trees)

If it’s not confidential, what type of problem would require manipulation and testing of trees?

Message board?

Almost all problems (that I find interesting): anytime you can frob two foos together and get a new foo (put two parts together to make something that can again be a part) you've got the material for a tree.

Concrete examples: Docker containers, PDF parts, POSIX filesystems, Scheme sources, SVGs, x86 objects, etc.

Do you have an example of doing this?

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