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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) 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".

Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122204 with the recursion scheme "SPINE", i.e. row n is obtained as SPINE(A057163 o ENIPS(A089840[n]) o A057163). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A130402. This table contains also all the rows of A122203 and A089840.

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122204 with the recursion scheme "FORK", or equivalently row n is obtained as FORK(ENIPS(n-th row of A089840)). See A122201 and A122204 for the description of FORK and ENIPS. Moreover, each row of A122287 can be obtained as the "DEEPEN" transform of the corresponding row in A122286. (See A122283 for the description of DEEPEN). Each row occurs only once in this table. Inverses of these permutations can be found in table A122288. This table contains also all the rows of A122201 and A089840.

Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122203 with the recursion scheme "ENIPS", i.e. row n is obtained as ENIPS(A057163 o SPINE(A089840[n]) o A057163). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A130403. This table contains also all the rows of A122204 and A089840.

Recursive transformation ENIPS for Catalan bijections has a well-defined inverse (see the definition & comments at A122204). For all Catalan bijections in A089840 that inverse produces a bijection which is itself in A089840. This sequence gives the indices to those positions where each ("primitive", non-recursive bijection) of A089840(n) occurs "atavistically" amongst the more complex recursive bijections in A122204. I.e. A122204(a(n)) = A089840(n). Similarly, other "atavistic forms" resurface as: A122287(a(n)) = A122201(n), A122286(a(n)) = A122203(n) and A122202(a(n)) = A122284(n). See also comments at A153833.

There exists similar atavistic index sequences computed for FORK (A122201) and KROF (A122202). Both start as 0,1654720,... (see A129604). This implies that regardless of how complex recursive derivations from A089840 one forms by repeatedly applying SPINE, ENIPS, FORK and/or KROF in some order (finite number of times), all the original primitive non-recursive elements of A089840 will eventually appear at some positions.

Other known terms: a(12)=65167, a(13)=65178, a(14)=65236, a(15)=169, a(16)=65302, a(22)-a(44) = 1656351, 1656576, 1656777, 1656628, 1656704, 1659507, 1659538, 1659653, 1659798, 1659685, 1659830, 1660155, 1660582, 1660439, 1660476, 1660621, 1660196, 1661073, 1660930, 1660859, 1661004, 1661287, 1661360.

This self-inverse automorphism is obtained as either SPINE(*A129608) or ENIPS(*A129608). See the definitions given in A122203 and A122204.