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This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) left subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153142. See further comments there and in A153141.

This bijection, Knack, is a ENIPS-transformation of the simple swap: ENIPS(*A069770) (i.e., row 1 of A122204). Furthermore, Knack and Knick (the inverse, A069767) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as row 1 in A122288 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knack)) = Knack.

Note: the name in Finnish is "Naks".

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) right subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153141. See further comments there.

This bijection, Knick, is a SPINE-transformation of the simple swap: SPINE(*A069770) (i.e., row 1 of A122203). Furthermore, Knick and Knack (the inverse, *A069768) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as a row 1 in A122287 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knick)) = Knick. There are also other peculiar properties.

Note: the name in Finnish is "Niks".

This sequence can be also obtained by starting complementing n's binary expansion from the second most significant bit, continuing towards lsb-end until the first 0-bit is reached, which is the last bit to be complemented.

In the Stern-Brocot enumeration system for positive rationals (A007305/A047679), this permutation converts the numerator into the denominator: A047679(n) = A007305(a(n)). - Yosu Yurramendi, Aug 30 2020

This permutation is induced by the third Lamplighter group generating wreath recursion a = s(b,a), b = (a,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 104 of Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end.

This permutation is induced by the first Lamplighter group generating wreath recursion a = s(a,b), b = (a,b) (i.e. binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 104 of Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end. It is the same automaton as given in figure 1 on page 211 of Grigorchuk and Zuk paper. Note that the fourth wreath recursion on page 104 of Bondarenko, et al. paper induces similarly the binary reflected Gray code A003188 (A054429-reflected conjugate of this permutation) and the second one induces Gray Code's inverse permutation A006068.

This permutation is induced by the Basilica group generating wreath recursion a = (1,b), b = s(1,a) (i.e. binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on the page 40 of Bartholdi and Virag paper, starting from the inactive (fixing) state a and rewriting bits from the second most significant bit to the least significant end.

This permutation of natural numbers is induced by the first generator of group 2861 mentioned on page 144 of "Classification of groups generated by 3-state automata over a 2-letter alphabet" paper. It can be computed by starting scanning n's binary expansion rightward from the second most significant bit, complementing every bit down to and including A) either the first 0-bit at even distance from the most significant bit or B) the first 1-bit at odd distance from the most significant bit.

Without the initial 0, a(n) is the lexicographically minimal sequence of distinct positive integers such that all values of a(n) mod n are distinct and nonnegative. - Ivan Neretin, Apr 27 2015

A002487(n)/A002487(n+1), n > 0, runs through all the reduced nonnegative rationals exactly once. A002487 is the Stern's sequence. Permutation from denominators (A002487(n+1))

1 2 1 3 2 3 1 4 3 5 2 5 3 4 1

where labels are

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

to numerators (A002487(n))

1 1 2 1 3 2 3 1 4 3 5 2 5 3 4

where changed labels are

1 3 2 7 4 5 6 15 8 9 10 11 12 13 14

Thus, b(n) = A002487(n+1), b(a(n)) = A002487(n), n>0. - Yosu Yurramendi, Jul 07 2016

These elements form a subgroup in A089840 (A089839) isomorphic to a group consisting of all finitely iterated wreath products of the form S_2 wr S_2 wr ... wr S_2 and each is an image of some finitary automorphism of an infinite binary tree. E.g. A089840(1) = *A069770 is an image of the generator A of Grigorchuk Group. See comments at A153246 and A153141.

The defining properties are propagated by all recursive transformations of A089840 which themselves do not behave differently depending whether there are internal or terminal vertices in the neighborhood of any vertex (at least the ones given in A122201-A122204, A122283-A122290, A130400-A130403), so this sequence gives also the corresponding positions in those tables.