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This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)

...B...C.......A...B

....\./.........\./

.A...x....-->....x...C.................A..().........()..A..

..\./.............\./...................\./....-->....\./...

...x...............x.....................x.............x....

(a . (b . c)) -> ((a . b) . c) ____ (a . ()) --> (() . a)

That is, we rotate the binary tree left, in case it is possible and otherwise (if the right hand side of a tree is a terminal node) swap the left and right subtree (so that the terminal node ends to the left hand side), i.e., apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.

This is the first multiclause nonrecursive automorphism in table A089840 and the first one whose order is not finite, i.e., the maximum size of cycles in this permutation is not bounded (see A089842). The cycle counts in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+1), which is otherwise the same sequence as for Catalan automorphisms *A057161/*A057162, but shifted once right. For an explanation, please see the notes in OEIS Wiki.

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)

A...B..............B...C

.\./................\./

..x...C..-->.....A...x................()..B.......B..()

...\./............\./..................\./...-->...\./.

....x..............x....................x...........x..

((a . b) . c) -> (a . (b . c)) __ (() . b) --> (b . ())

That is, we rotate the binary tree right, in case it is possible and otherwise (if the left hand side of a tree is a terminal node) swap the right and left subtree (so that the terminal node ends to the right hand side), i.e. apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.

See also the comments at A074679.

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).

.....B...C.......C...A

......\./.........\./

...A...x...-->... .x...B...............A..().........()..A..

....\./.............\./.................\./....-->....\./...

.....x...............x...................x.............x....

(a . (b . c)) --> ((c . a) . b) ___ (a . ()) --> (() . a)

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).

.A...B...............C...A

..\./.................\./

...x...C...-->.....B...x...............()..A.........A..()..

....\./.............\./.................\./....-->....\./...

.....x...............x...................x.............x....

((a . b) . c) --> (b . (c . a)) __ (() . a) ----> (a . ())

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

This bijection of binary trees is obtained when we apply bijection *A074679 to the left subtree and keep the right subtree intact.

....B...C.......A...B

.....\./.........\./

..A...x....-->....x...C.................A..().........()..A....

...\./.............\./...................\./....-->....\./.....

....x...D...........x...D.................x...C.........x...C..

.....\./.............\./...................\./...........\./...

......x...............x.....................x.............x....

...............................................................

Compare to A154121.

See "Catalan Automorphisms" OEIS-Wiki page for a detailed explanation how to obtain a given integer sequence from this definition.

This bijection of binary trees is obtained when we apply bijection *A074680 to the left subtree and keep the right subtree intact.

.A...B...............B...C

..\./.................\./

...x...C....-->....A...x.................()..B.........B...()....

....\./.............\./...................\./....-->....\./.....

.....x...D...........x...D.................x...C.........x...C..

......\./.............\./...................\./...........\./...

.......x...............x.....................x.............x....

................................................................

Compare to A154122.

See "Catalan bijections" OEIS-Wiki page for a detailed explanation how to obtain a given integer sequence from this definition.

If a(n) is nonzero, then the n-th non-recursive Catalan Automorphism in A089840 does not have orbits (cycles) larger than that and the corresponding LCM-sequence will set to a constant sequence a(n),a(n),a(n),a(n),... E.g. A089840[4] = A089851 is obtained by rotating three subtrees cyclically and its LCM-sequence begins as 1,1,1,3,3,3,3,3,3,3,3,... (a(4)=3). Similarly, A089840[15] = A089859, whose LCM-sequence begins as 1,1,2,4,4,4,4,4,4,4,4,.... (a(15)=4). In contrast, the Max. cycle and LCM-sequence (A089410) of A089840[12] (= A074679) exhibits genuine growth, thus a(12)=0.