- FindStat

Identifier

Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths

Images

[] => [] => []

[[]] => [[]] => [1,0]

[[],[]] => [[[]]] => [1,1,0,0]

[[[]]] => [[],[]] => [1,0,1,0]

[[],[],[]] => [[[[]]]] => [1,1,1,0,0,0]

[[],[[]]] => [[[],[]]] => [1,1,0,1,0,0]

[[[]],[]] => [[],[[]]] => [1,0,1,1,0,0]

[[[],[]]] => [[[]],[]] => [1,1,0,0,1,0]

[[[[]]]] => [[],[],[]] => [1,0,1,0,1,0]

[[],[],[],[]] => [[[[[]]]]] => [1,1,1,1,0,0,0,0]

[[],[],[[]]] => [[[[],[]]]] => [1,1,1,0,1,0,0,0]

[[],[[]],[]] => [[[],[[]]]] => [1,1,0,1,1,0,0,0]

[[],[[],[]]] => [[[[]],[]]] => [1,1,1,0,0,1,0,0]

[[],[[[]]]] => [[[],[],[]]] => [1,1,0,1,0,1,0,0]

[[[]],[],[]] => [[],[[[]]]] => [1,0,1,1,1,0,0,0]

[[[]],[[]]] => [[],[[],[]]] => [1,0,1,1,0,1,0,0]

[[[],[]],[]] => [[[]],[[]]] => [1,1,0,0,1,1,0,0]

[[[[]]],[]] => [[],[],[[]]] => [1,0,1,0,1,1,0,0]

[[[],[],[]]] => [[[[]]],[]] => [1,1,1,0,0,0,1,0]

[[[],[[]]]] => [[[],[]],[]] => [1,1,0,1,0,0,1,0]

[[[[]],[]]] => [[],[[]],[]] => [1,0,1,1,0,0,1,0]

[[[[],[]]]] => [[[]],[],[]] => [1,1,0,0,1,0,1,0]

[[[[[]]]]] => [[],[],[],[]] => [1,0,1,0,1,0,1,0]

[[],[],[],[],[]] => [[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0]

[[],[],[],[[]]] => [[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0]

[[],[],[[]],[]] => [[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0]

[[],[],[[],[]]] => [[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0]

[[],[],[[[]]]] => [[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0]

[[],[[]],[],[]] => [[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0]

[[],[[]],[[]]] => [[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0]

[[],[[],[]],[]] => [[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0]

[[],[[[]]],[]] => [[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0]

[[],[[],[],[]]] => [[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0]

[[],[[],[[]]]] => [[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0]

[[],[[[]],[]]] => [[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0]

[[],[[[],[]]]] => [[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0]

[[],[[[[]]]]] => [[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0]

[[[]],[],[],[]] => [[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0]

[[[]],[],[[]]] => [[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0]

[[[]],[[]],[]] => [[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0]

[[[]],[[],[]]] => [[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0]

[[[]],[[[]]]] => [[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0]

[[[],[]],[],[]] => [[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0]

[[[[]]],[],[]] => [[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0]

[[[],[]],[[]]] => [[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0]

[[[[]]],[[]]] => [[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0]

[[[],[],[]],[]] => [[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0]

[[[],[[]]],[]] => [[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0]

[[[[]],[]],[]] => [[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0]

[[[[],[]]],[]] => [[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0]

[[[[[]]]],[]] => [[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0]

[[[],[],[],[]]] => [[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0]

[[[],[],[[]]]] => [[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0]

[[[],[[]],[]]] => [[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0]

[[[],[[],[]]]] => [[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0]

[[[],[[[]]]]] => [[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0]

[[[[]],[],[]]] => [[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0]

[[[[]],[[]]]] => [[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0]

[[[[],[]],[]]] => [[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0]

[[[[[]]],[]]] => [[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0]

[[[[],[],[]]]] => [[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0]

[[[[],[[]]]]] => [[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0]

[[[[[]],[]]]] => [[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0]

[[[[[],[]]]]] => [[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0]

[[[[[[]]]]]] => [[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0]

[[],[],[],[],[],[]] => [[[[[[[]]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0]

[[],[],[],[],[[]]] => [[[[[[],[]]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0]

[[],[],[],[[]],[]] => [[[[[],[[]]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0]

[[],[],[],[[],[]]] => [[[[[[]],[]]]]] => [1,1,1,1,1,0,0,1,0,0,0,0]

[[],[],[],[[[]]]] => [[[[[],[],[]]]]] => [1,1,1,1,0,1,0,1,0,0,0,0]

[[],[],[[]],[],[]] => [[[[],[[[]]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0]

[[],[],[[]],[[]]] => [[[[],[[],[]]]]] => [1,1,1,0,1,1,0,1,0,0,0,0]

[[],[],[[],[]],[]] => [[[[[]],[[]]]]] => [1,1,1,1,0,0,1,1,0,0,0,0]

[[],[],[[[]]],[]] => [[[[],[],[[]]]]] => [1,1,1,0,1,0,1,1,0,0,0,0]

[[],[],[[],[],[]]] => [[[[[[]]],[]]]] => [1,1,1,1,1,0,0,0,1,0,0,0]

[[],[],[[],[[]]]] => [[[[[],[]],[]]]] => [1,1,1,1,0,1,0,0,1,0,0,0]

[[],[],[[[]],[]]] => [[[[],[[]],[]]]] => [1,1,1,0,1,1,0,0,1,0,0,0]

[[],[],[[[],[]]]] => [[[[[]],[],[]]]] => [1,1,1,1,0,0,1,0,1,0,0,0]

[[],[],[[[[]]]]] => [[[[],[],[],[]]]] => [1,1,1,0,1,0,1,0,1,0,0,0]

[[],[[]],[],[],[]] => [[[],[[[[]]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0]

[[],[[]],[],[[]]] => [[[],[[[],[]]]]] => [1,1,0,1,1,1,0,1,0,0,0,0]

[[],[[]],[[]],[]] => [[[],[[],[[]]]]] => [1,1,0,1,1,0,1,1,0,0,0,0]

[[],[[]],[[],[]]] => [[[],[[[]],[]]]] => [1,1,0,1,1,1,0,0,1,0,0,0]

[[],[[]],[[[]]]] => [[[],[[],[],[]]]] => [1,1,0,1,1,0,1,0,1,0,0,0]

[[],[[],[]],[],[]] => [[[[]],[[[]]]]] => [1,1,1,0,0,1,1,1,0,0,0,0]

[[],[[[]]],[],[]] => [[[],[],[[[]]]]] => [1,1,0,1,0,1,1,1,0,0,0,0]

[[],[[],[]],[[]]] => [[[[]],[[],[]]]] => [1,1,1,0,0,1,1,0,1,0,0,0]

[[],[[[]]],[[]]] => [[[],[],[[],[]]]] => [1,1,0,1,0,1,1,0,1,0,0,0]

[[],[[],[],[]],[]] => [[[[[]]],[[]]]] => [1,1,1,1,0,0,0,1,1,0,0,0]

[[],[[],[[]]],[]] => [[[[],[]],[[]]]] => [1,1,1,0,1,0,0,1,1,0,0,0]

[[],[[[]],[]],[]] => [[[],[[]],[[]]]] => [1,1,0,1,1,0,0,1,1,0,0,0]

[[],[[[],[]]],[]] => [[[[]],[],[[]]]] => [1,1,1,0,0,1,0,1,1,0,0,0]

[[],[[[[]]]],[]] => [[[],[],[],[[]]]] => [1,1,0,1,0,1,0,1,1,0,0,0]

[[],[[],[],[],[]]] => [[[[[[]]]],[]]] => [1,1,1,1,1,0,0,0,0,1,0,0]

[[],[[],[],[[]]]] => [[[[[],[]]],[]]] => [1,1,1,1,0,1,0,0,0,1,0,0]

[[],[[],[[]],[]]] => [[[[],[[]]],[]]] => [1,1,1,0,1,1,0,0,0,1,0,0]

[[],[[],[[],[]]]] => [[[[[]],[]],[]]] => [1,1,1,1,0,0,1,0,0,1,0,0]

[[],[[],[[[]]]]] => [[[[],[],[]],[]]] => [1,1,1,0,1,0,1,0,0,1,0,0]

[[],[[[]],[],[]]] => [[[],[[[]]],[]]] => [1,1,0,1,1,1,0,0,0,1,0,0]

[[],[[[]],[[]]]] => [[[],[[],[]],[]]] => [1,1,0,1,1,0,1,0,0,1,0,0]

[[],[[[],[]],[]]] => [[[[]],[[]],[]]] => [1,1,1,0,0,1,1,0,0,1,0,0]

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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

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.