Three more things about Beren's tree

1. The description in terms of even/odd number of zeroes doesn't have to be proven via dual Zeckendorf; it can also be proven directly by Ophelia's cold/hot argument, rephrased for this purpose.

2. Speaking of which -- one could define an analogous tree for dual Zeckendorf starting from the empty string. I don't know of any way to relate that to any of these other trees, though, nor am I aware of its reading permutation or its inverse being anything interesting.

(There's one almost-match for the inverse on OEIS, namely A303765, but it doesn't match. Not unless I've seriously messed up.)

3. I put up a page about it on my website!

EDIT: Ugh, the tree totally does relate after all. Gonna need to go back and edit this page...

EDIT: OK, did a quick edit to account for the new stuff... go just read about it there, I guess! (There's a new parity-based tree. :P )

EDIT again next day: OMG there's 8 more trees I missed. Well now they're in there too, which meant some rewriting. Geez. Yay, more ASCII art.