A057163-conjugation
We say that the gatomorphism g is the A057163-conjugate of the gatomorphism f if g = gmA057163 o f o gmA057163, that is, we reflect the argument as a binary tree before feeding it to f, and then reflect again its result (as a binary tree).



Embeddability

Definition. We say that
gatomorphism f embeds into gatomorphism g in scale n:m
if there is an injection:
e: Catn (parenthesizations/trees/etc. of size n)  -->  Catm (parenthesizations/trees/etc. of size m)
that for all the elements a of the former set, and for all integer values of the superscript r in Z (which stands for the repeated applications of f or g (if the exponent is positive) or their inverse (if the exponent is negative)) it holds that
(gr (e a)) = (e (fr a))


Examples.

Definition. We say that
gatomorphism g is Lukasiewicz-word permuting
if the Lukasiewicz-word of (g s) is always a permutation of the Lukasiewicz-word of s for all trees/parenthesizations s of any size.

This implies that the subset of the planar binary trees (of the general trees) is closed under the action of g, and thus its restriction to that subset induces another gatomorphism f. Rephrasing this in above terms, we can say that gatomorphism f embeds into gatomorphism g in scale n:2n.


Definition. We say that
gatomorphism g is self-embeddable
if g embeds into itself in scale n:m, where n ranges through all values in N, and m is a function of n, with m > n.

Two special cases of self-embeddability deserve special attention:

Definition. We say that

gatomorphism g is "horizontally telescoping"
if for all the parenthesizations s it holds that
(g (cons '( ) s)) = (cons '( ) (g  s))
where the injection e required by our general definition of embeddability is now defined as
(define (e s)  (cons '( ) s))
which maps the parenthesizations/trees of size n to the parenthesizations/trees of size n+1 by simply concatenating empty parentheses to the front of the corresponding parenthesization, or in terms of binary trees, by grafting an old tree of size n as a right-hand-side subtree of a new tree of size n+1 whose left-hand-side subtree is just a single edge \.

Remark. This is equivalent to saying that each sub-permutation of the length A000108(n) induced by g's action on the standard sequence of lexicographically ordered parenthesizations A014486 (A063171) starts with the same cycle-structure as the previous sub-permutation. In this case we can form yet another permutation of the natural numbers by conceptually taking the "infiniteth" of such sub-permutations and by "normalizing" it to begin from 0 or 1.
For example, Meeussen's breadth-first <-> depth-first conversion for binary trees A057117 satisfies the above condition, and by thus "contracting the telescope" we get another permutation of natural numbers, A038776.

Definition. We say that

gatomorphism g is "vertically telescoping"
if for all parenthesizations s it holds that
(g (cons s '( ))) is equal to (cons (g s) '( ))
where the injection e is now defined as
(define (e s)  (cons s '( )))
which maps the parenthesizations/trees of size n to the parenthesizations/trees of size n+1 by surrounding the corresponding parenthesization with one extra parentheses ( and ), or in terms of binary trees, by grafting an old tree of size n as a left-hand-side subtree of a new tree of size n+1 whose right-hand-side subtree is just a single edge /.

Remark. For example, it is easy to see that this holds for the gatomorphism A057164 (Deep Reverse), and similarly for A057511/A057512 (Deep Rotates), and as A069787 = A057163 o A057164 o A057163, we can contract A069787 (a horizontally telescoping gatomorphism) to get A072799.


Remark. The A057163-conjugate of any "horizontally telescoping" gatomorphism is "vertically telescoping" and vice versa. Like with all self-embeddable gatomorphisms of the scale n:n+1 we can be sure that the corresponding cycle/orbit count sequences of these gatomorphisms are genuinely monotone (growing) after size n>0. This is because one can find the same cycles/orbits from the set of Catn+1 parenthesizations/trees/whatever of size n+1 as what one finds in the set of Catn parenthesizations/trees/whatever of size n and the remaining Catn+1 - Catn parenthesizations/trees/whatever must form at least one extra orbit.

Remark. The allusions "horizontal" and "vertical" should be obvious if one thinks in the terms of Dyck paths (Catalan Mountain Ranges), and also the plane general trees in the latter case.