This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant A quaternary encoding" as explained in the sequence A085184.

This is a rare example of a simply defined Catalan Automorphism where the cycle count sequence (A089411) is not monotone. (See A127296 for a much more complex example.)

The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in subpermutations limited by A014137 and A014138 are given by A089411, A086586 and A089412.