- FindStat

Identifier

Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
Mp00246: Ordered trees —rotate⟶ Ordered trees
St000415: Ordered trees ⟶ ℤ

Values

[[]] => [[]] => [[]] => 1

[[],[]] => [[[]]] => [[[]]] => 1

[[[]]] => [[],[]] => [[],[]] => 2

[[],[],[]] => [[[[]]]] => [[[[]]]] => 1

[[],[[]]] => [[[],[]]] => [[],[[]]] => 2

[[[]],[]] => [[],[[]]] => [[[]],[]] => 2

[[[],[]]] => [[[]],[]] => [[[],[]]] => 2

[[[[]]]] => [[],[],[]] => [[],[],[]] => 6

[[],[],[],[]] => [[[[[]]]]] => [[[[[]]]]] => 1

[[],[],[[]]] => [[[[],[]]]] => [[],[[[]]]] => 2

[[],[[]],[]] => [[[],[[]]]] => [[[]],[[]]] => 2

[[],[[],[]]] => [[[[]],[]]] => [[[],[[]]]] => 2

[[],[[[]]]] => [[[],[],[]]] => [[],[],[[]]] => 6

[[[]],[],[]] => [[],[[[]]]] => [[[[]]],[]] => 2

[[[]],[[]]] => [[],[[],[]]] => [[[],[]],[]] => 4

[[[],[]],[]] => [[[]],[[]]] => [[[[]],[]]] => 2

[[[[]]],[]] => [[],[],[[]]] => [[],[[]],[]] => 6

[[[],[],[]]] => [[[[]]],[]] => [[[[],[]]]] => 2

[[[],[[]]]] => [[[],[]],[]] => [[],[[],[]]] => 4

[[[[]],[]]] => [[],[[]],[]] => [[[]],[],[]] => 6

[[[[],[]]]] => [[[]],[],[]] => [[[],[],[]]] => 6

[[[[[]]]]] => [[],[],[],[]] => [[],[],[],[]] => 24

[[],[],[],[],[]] => [[[[[[]]]]]] => [[[[[[]]]]]] => 1

[[],[],[],[[]]] => [[[[[],[]]]]] => [[],[[[[]]]]] => 2

[[],[],[[]],[]] => [[[[],[[]]]]] => [[[]],[[[]]]] => 2

[[],[],[[],[]]] => [[[[[]],[]]]] => [[[],[[[]]]]] => 2

[[],[],[[[]]]] => [[[[],[],[]]]] => [[],[],[[[]]]] => 6

[[],[[]],[],[]] => [[[],[[[]]]]] => [[[[]]],[[]]] => 2

[[],[[]],[[]]] => [[[],[[],[]]]] => [[[],[]],[[]]] => 4

[[],[[],[]],[]] => [[[[]],[[]]]] => [[[[]],[[]]]] => 2

[[],[[[]]],[]] => [[[],[],[[]]]] => [[],[[]],[[]]] => 6

[[],[[],[],[]]] => [[[[[]]],[]]] => [[[[],[[]]]]] => 2

[[],[[],[[]]]] => [[[[],[]],[]]] => [[],[[],[[]]]] => 4

[[],[[[]],[]]] => [[[],[[]],[]]] => [[[]],[],[[]]] => 6

[[],[[[],[]]]] => [[[[]],[],[]]] => [[[],[],[[]]]] => 6

[[],[[[[]]]]] => [[[],[],[],[]]] => [[],[],[],[[]]] => 24

[[[]],[],[],[]] => [[],[[[[]]]]] => [[[[[]]]],[]] => 2

[[[]],[],[[]]] => [[],[[[],[]]]] => [[[[],[]]],[]] => 4

[[[]],[[]],[]] => [[],[[],[[]]]] => [[[],[[]]],[]] => 4

[[[]],[[],[]]] => [[],[[[]],[]]] => [[[[]],[]],[]] => 4

[[[]],[[[]]]] => [[],[[],[],[]]] => [[[],[],[]],[]] => 12

[[[],[]],[],[]] => [[[]],[[[]]]] => [[[[[]]],[]]] => 2

[[[[]]],[],[]] => [[],[],[[[]]]] => [[],[[[]]],[]] => 6

[[[],[]],[[]]] => [[[]],[[],[]]] => [[[[],[]],[]]] => 4

[[[[]]],[[]]] => [[],[],[[],[]]] => [[],[[],[]],[]] => 12

[[[],[],[]],[]] => [[[[]]],[[]]] => [[[[[]],[]]]] => 2

[[[],[[]]],[]] => [[[],[]],[[]]] => [[],[[[]],[]]] => 4

[[[[]],[]],[]] => [[],[[]],[[]]] => [[[]],[[]],[]] => 6

[[[[],[]]],[]] => [[[]],[],[[]]] => [[[],[[]],[]]] => 6

[[[[[]]]],[]] => [[],[],[],[[]]] => [[],[],[[]],[]] => 24

[[[],[],[],[]]] => [[[[[]]]],[]] => [[[[[],[]]]]] => 2

[[[],[],[[]]]] => [[[[],[]]],[]] => [[],[[[],[]]]] => 4

[[[],[[]],[]]] => [[[],[[]]],[]] => [[[]],[[],[]]] => 4

[[[],[[],[]]]] => [[[[]],[]],[]] => [[[],[[],[]]]] => 4

[[[],[[[]]]]] => [[[],[],[]],[]] => [[],[],[[],[]]] => 12

[[[[]],[],[]]] => [[],[[[]]],[]] => [[[[]]],[],[]] => 6

[[[[]],[[]]]] => [[],[[],[]],[]] => [[[],[]],[],[]] => 12

[[[[],[]],[]]] => [[[]],[[]],[]] => [[[[]],[],[]]] => 6

[[[[[]]],[]]] => [[],[],[[]],[]] => [[],[[]],[],[]] => 24

[[[[],[],[]]]] => [[[[]]],[],[]] => [[[[],[],[]]]] => 6

[[[[],[[]]]]] => [[[],[]],[],[]] => [[],[[],[],[]]] => 12

[[[[[]],[]]]] => [[],[[]],[],[]] => [[[]],[],[],[]] => 24

[[[[[],[]]]]] => [[[]],[],[],[]] => [[[],[],[],[]]] => 24

[[[[[[]]]]]] => [[],[],[],[],[]] => [[],[],[],[],[]] => 120

[[],[],[],[],[],[]] => [[[[[[[]]]]]]] => [[[[[[[]]]]]]] => 1

[[],[],[],[],[[]]] => [[[[[[],[]]]]]] => [[],[[[[[]]]]]] => 2

[[],[],[],[[]],[]] => [[[[[],[[]]]]]] => [[[]],[[[[]]]]] => 2

[[],[],[],[[],[]]] => [[[[[[]],[]]]]] => [[[],[[[[]]]]]] => 2

[[],[],[],[[[]]]] => [[[[[],[],[]]]]] => [[],[],[[[[]]]]] => 6

[[],[],[[]],[],[]] => [[[[],[[[]]]]]] => [[[[]]],[[[]]]] => 2

[[],[],[[]],[[]]] => [[[[],[[],[]]]]] => [[[],[]],[[[]]]] => 4

[[],[],[[],[]],[]] => [[[[[]],[[]]]]] => [[[[]],[[[]]]]] => 2

[[],[],[[[]]],[]] => [[[[],[],[[]]]]] => [[],[[]],[[[]]]] => 6

[[],[],[[],[],[]]] => [[[[[[]]],[]]]] => [[[[],[[[]]]]]] => 2

[[],[],[[],[[]]]] => [[[[[],[]],[]]]] => [[],[[],[[[]]]]] => 4

[[],[],[[[]],[]]] => [[[[],[[]],[]]]] => [[[]],[],[[[]]]] => 6

[[],[],[[[],[]]]] => [[[[[]],[],[]]]] => [[[],[],[[[]]]]] => 6

[[],[],[[[[]]]]] => [[[[],[],[],[]]]] => [[],[],[],[[[]]]] => 24

[[],[[]],[],[],[]] => [[[],[[[[]]]]]] => [[[[[]]]],[[]]] => 2

[[],[[]],[],[[]]] => [[[],[[[],[]]]]] => [[[[],[]]],[[]]] => 4

[[],[[]],[[]],[]] => [[[],[[],[[]]]]] => [[[],[[]]],[[]]] => 4

[[],[[]],[[],[]]] => [[[],[[[]],[]]]] => [[[[]],[]],[[]]] => 4

[[],[[]],[[[]]]] => [[[],[[],[],[]]]] => [[[],[],[]],[[]]] => 12

[[],[[],[]],[],[]] => [[[[]],[[[]]]]] => [[[[[]]],[[]]]] => 2

[[],[[[]]],[],[]] => [[[],[],[[[]]]]] => [[],[[[]]],[[]]] => 6

[[],[[],[]],[[]]] => [[[[]],[[],[]]]] => [[[[],[]],[[]]]] => 4

[[],[[[]]],[[]]] => [[[],[],[[],[]]]] => [[],[[],[]],[[]]] => 12

[[],[[],[],[]],[]] => [[[[[]]],[[]]]] => [[[[[]],[[]]]]] => 2

[[],[[],[[]]],[]] => [[[[],[]],[[]]]] => [[],[[[]],[[]]]] => 4

[[],[[[]],[]],[]] => [[[],[[]],[[]]]] => [[[]],[[]],[[]]] => 6

[[],[[[],[]]],[]] => [[[[]],[],[[]]]] => [[[],[[]],[[]]]] => 6

[[],[[[[]]]],[]] => [[[],[],[],[[]]]] => [[],[],[[]],[[]]] => 24

[[],[[],[],[],[]]] => [[[[[[]]]],[]]] => [[[[[],[[]]]]]] => 2

[[],[[],[],[[]]]] => [[[[[],[]]],[]]] => [[],[[[],[[]]]]] => 4

[[],[[],[[]],[]]] => [[[[],[[]]],[]]] => [[[]],[[],[[]]]] => 4

[[],[[],[[],[]]]] => [[[[[]],[]],[]]] => [[[],[[],[[]]]]] => 4

[[],[[],[[[]]]]] => [[[[],[],[]],[]]] => [[],[],[[],[[]]]] => 12

[[],[[[]],[],[]]] => [[[],[[[]]],[]]] => [[[[]]],[],[[]]] => 6

[[],[[[]],[[]]]] => [[[],[[],[]],[]]] => [[[],[]],[],[[]]] => 12

[[],[[[],[]],[]]] => [[[[]],[[]],[]]] => [[[[]],[],[[]]]] => 6

[[],[[[[]]],[]]] => [[[],[],[[]],[]]] => [[],[[]],[],[[]]] => 24

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Description

The size of the automorphism group of the rooted tree underlying the ordered tree.

Map

DeBruijn-Morselt plane tree automorphism

Description

This automorphism on the set of plane trees with a given number of vertices is the combination of the two "canonical" bijections between plane trees and binary trees, using either the left or right branches. In [3] Shapiro essentially attributes this automorphism to DeBruijn and Morselt [1], hence the name I chose. Shapiro shows in that paper that for a large subset of plane trees (those corresponding to compositions), the sixth power of the automorphism acts as the identity on this subset. However, its behavior in general is quite chaotic. In [2] Donaghey studies this automorphism further and shows that its behavior on all trees can be reduced to its behavior on certain "primitive" trees.

Map

rotate

Description

The rotation of an ordered tree.
Let $T$ be a rooted ordered tree with root $r$ and left most leaf $\ell$. It's rotation is obtained by attaching a leaf to $r$, removing the leaf $\ell$ and letting the vertex attached to $\ell$ be the new root.
By duality, this operation corresponds to rotation of polygon dissections.