- FindStat

Identifier

Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees

Images

[.,.] => [[]] => [[]]

[.,[.,.]] => [[],[]] => [[[]]]

[[.,.],.] => [[[]]] => [[],[]]

[.,[.,[.,.]]] => [[],[],[]] => [[[[]]]]

[.,[[.,.],.]] => [[],[[]]] => [[[],[]]]

[[.,.],[.,.]] => [[[]],[]] => [[],[[]]]

[[.,[.,.]],.] => [[[],[]]] => [[[]],[]]

[[[.,.],.],.] => [[[[]]]] => [[],[],[]]

[.,[.,[.,[.,.]]]] => [[],[],[],[]] => [[[[[]]]]]

[.,[.,[[.,.],.]]] => [[],[],[[]]] => [[[[],[]]]]

[.,[[.,.],[.,.]]] => [[],[[]],[]] => [[[],[[]]]]

[.,[[.,[.,.]],.]] => [[],[[],[]]] => [[[[]],[]]]

[.,[[[.,.],.],.]] => [[],[[[]]]] => [[[],[],[]]]

[[.,.],[.,[.,.]]] => [[[]],[],[]] => [[],[[[]]]]

[[.,.],[[.,.],.]] => [[[]],[[]]] => [[],[[],[]]]

[[.,[.,.]],[.,.]] => [[[],[]],[]] => [[[]],[[]]]

[[[.,.],.],[.,.]] => [[[[]]],[]] => [[],[],[[]]]

[[.,[.,[.,.]]],.] => [[[],[],[]]] => [[[[]]],[]]

[[.,[[.,.],.]],.] => [[[],[[]]]] => [[[],[]],[]]

[[[.,.],[.,.]],.] => [[[[]],[]]] => [[],[[]],[]]

[[[.,[.,.]],.],.] => [[[[],[]]]] => [[[]],[],[]]

[[[[.,.],.],.],.] => [[[[[]]]]] => [[],[],[],[]]

[.,[.,[.,[.,[.,.]]]]] => [[],[],[],[],[]] => [[[[[[]]]]]]

[.,[.,[.,[[.,.],.]]]] => [[],[],[],[[]]] => [[[[[],[]]]]]

[.,[.,[[.,.],[.,.]]]] => [[],[],[[]],[]] => [[[[],[[]]]]]

[.,[.,[[.,[.,.]],.]]] => [[],[],[[],[]]] => [[[[[]],[]]]]

[.,[.,[[[.,.],.],.]]] => [[],[],[[[]]]] => [[[[],[],[]]]]

[.,[[.,.],[.,[.,.]]]] => [[],[[]],[],[]] => [[[],[[[]]]]]

[.,[[.,.],[[.,.],.]]] => [[],[[]],[[]]] => [[[],[[],[]]]]

[.,[[.,[.,.]],[.,.]]] => [[],[[],[]],[]] => [[[[]],[[]]]]

[.,[[[.,.],.],[.,.]]] => [[],[[[]]],[]] => [[[],[],[[]]]]

[.,[[.,[.,[.,.]]],.]] => [[],[[],[],[]]] => [[[[[]]],[]]]

[.,[[.,[[.,.],.]],.]] => [[],[[],[[]]]] => [[[[],[]],[]]]

[.,[[[.,.],[.,.]],.]] => [[],[[[]],[]]] => [[[],[[]],[]]]

[.,[[[.,[.,.]],.],.]] => [[],[[[],[]]]] => [[[[]],[],[]]]

[.,[[[[.,.],.],.],.]] => [[],[[[[]]]]] => [[[],[],[],[]]]

[[.,.],[.,[.,[.,.]]]] => [[[]],[],[],[]] => [[],[[[[]]]]]

[[.,.],[.,[[.,.],.]]] => [[[]],[],[[]]] => [[],[[[],[]]]]

[[.,.],[[.,.],[.,.]]] => [[[]],[[]],[]] => [[],[[],[[]]]]

[[.,.],[[.,[.,.]],.]] => [[[]],[[],[]]] => [[],[[[]],[]]]

[[.,.],[[[.,.],.],.]] => [[[]],[[[]]]] => [[],[[],[],[]]]

[[.,[.,.]],[.,[.,.]]] => [[[],[]],[],[]] => [[[]],[[[]]]]

[[.,[.,.]],[[.,.],.]] => [[[],[]],[[]]] => [[[]],[[],[]]]

[[[.,.],.],[.,[.,.]]] => [[[[]]],[],[]] => [[],[],[[[]]]]

[[[.,.],.],[[.,.],.]] => [[[[]]],[[]]] => [[],[],[[],[]]]

[[.,[.,[.,.]]],[.,.]] => [[[],[],[]],[]] => [[[[]]],[[]]]

[[.,[[.,.],.]],[.,.]] => [[[],[[]]],[]] => [[[],[]],[[]]]

[[[.,.],[.,.]],[.,.]] => [[[[]],[]],[]] => [[],[[]],[[]]]

[[[.,[.,.]],.],[.,.]] => [[[[],[]]],[]] => [[[]],[],[[]]]

[[[[.,.],.],.],[.,.]] => [[[[[]]]],[]] => [[],[],[],[[]]]

[[.,[.,[.,[.,.]]]],.] => [[[],[],[],[]]] => [[[[[]]]],[]]

[[.,[.,[[.,.],.]]],.] => [[[],[],[[]]]] => [[[[],[]]],[]]

[[.,[[.,.],[.,.]]],.] => [[[],[[]],[]]] => [[[],[[]]],[]]

[[.,[[.,[.,.]],.]],.] => [[[],[[],[]]]] => [[[[]],[]],[]]

[[.,[[[.,.],.],.]],.] => [[[],[[[]]]]] => [[[],[],[]],[]]

[[[.,.],[.,[.,.]]],.] => [[[[]],[],[]]] => [[],[[[]]],[]]

[[[.,.],[[.,.],.]],.] => [[[[]],[[]]]] => [[],[[],[]],[]]

[[[.,[.,.]],[.,.]],.] => [[[[],[]],[]]] => [[[]],[[]],[]]

[[[[.,.],.],[.,.]],.] => [[[[[]]],[]]] => [[],[],[[]],[]]

[[[.,[.,[.,.]]],.],.] => [[[[],[],[]]]] => [[[[]]],[],[]]

[[[.,[[.,.],.]],.],.] => [[[[],[[]]]]] => [[[],[]],[],[]]

[[[[.,.],[.,.]],.],.] => [[[[[]],[]]]] => [[],[[]],[],[]]

[[[[.,[.,.]],.],.],.] => [[[[[],[]]]]] => [[[]],[],[],[]]

[[[[[.,.],.],.],.],.] => [[[[[[]]]]]] => [[],[],[],[],[]]

[.,[.,[.,[.,[.,[.,.]]]]]] => [[],[],[],[],[],[]] => [[[[[[[]]]]]]]

[.,[.,[.,[.,[[.,.],.]]]]] => [[],[],[],[],[[]]] => [[[[[[],[]]]]]]

[.,[.,[.,[[.,.],[.,.]]]]] => [[],[],[],[[]],[]] => [[[[[],[[]]]]]]

[.,[.,[.,[[.,[.,.]],.]]]] => [[],[],[],[[],[]]] => [[[[[[]],[]]]]]

[.,[.,[.,[[[.,.],.],.]]]] => [[],[],[],[[[]]]] => [[[[[],[],[]]]]]

[.,[.,[[.,.],[.,[.,.]]]]] => [[],[],[[]],[],[]] => [[[[],[[[]]]]]]

[.,[.,[[.,.],[[.,.],.]]]] => [[],[],[[]],[[]]] => [[[[],[[],[]]]]]

[.,[.,[[.,[.,.]],[.,.]]]] => [[],[],[[],[]],[]] => [[[[[]],[[]]]]]

[.,[.,[[[.,.],.],[.,.]]]] => [[],[],[[[]]],[]] => [[[[],[],[[]]]]]

[.,[.,[[.,[.,[.,.]]],.]]] => [[],[],[[],[],[]]] => [[[[[[]]],[]]]]

[.,[.,[[.,[[.,.],.]],.]]] => [[],[],[[],[[]]]] => [[[[[],[]],[]]]]

[.,[.,[[[.,.],[.,.]],.]]] => [[],[],[[[]],[]]] => [[[[],[[]],[]]]]

[.,[.,[[[.,[.,.]],.],.]]] => [[],[],[[[],[]]]] => [[[[[]],[],[]]]]

[.,[.,[[[[.,.],.],.],.]]] => [[],[],[[[[]]]]] => [[[[],[],[],[]]]]

[.,[[.,.],[.,[.,[.,.]]]]] => [[],[[]],[],[],[]] => [[[],[[[[]]]]]]

[.,[[.,.],[.,[[.,.],.]]]] => [[],[[]],[],[[]]] => [[[],[[[],[]]]]]

[.,[[.,.],[[.,.],[.,.]]]] => [[],[[]],[[]],[]] => [[[],[[],[[]]]]]

[.,[[.,.],[[.,[.,.]],.]]] => [[],[[]],[[],[]]] => [[[],[[[]],[]]]]

[.,[[.,.],[[[.,.],.],.]]] => [[],[[]],[[[]]]] => [[[],[[],[],[]]]]

[.,[[.,[.,.]],[.,[.,.]]]] => [[],[[],[]],[],[]] => [[[[]],[[[]]]]]

[.,[[.,[.,.]],[[.,.],.]]] => [[],[[],[]],[[]]] => [[[[]],[[],[]]]]

[.,[[[.,.],.],[.,[.,.]]]] => [[],[[[]]],[],[]] => [[[],[],[[[]]]]]

[.,[[[.,.],.],[[.,.],.]]] => [[],[[[]]],[[]]] => [[[],[],[[],[]]]]

[.,[[.,[.,[.,.]]],[.,.]]] => [[],[[],[],[]],[]] => [[[[[]]],[[]]]]

[.,[[.,[[.,.],.]],[.,.]]] => [[],[[],[[]]],[]] => [[[[],[]],[[]]]]

[.,[[[.,.],[.,.]],[.,.]]] => [[],[[[]],[]],[]] => [[[],[[]],[[]]]]

[.,[[[.,[.,.]],.],[.,.]]] => [[],[[[],[]]],[]] => [[[[]],[],[[]]]]

[.,[[[[.,.],.],.],[.,.]]] => [[],[[[[]]]],[]] => [[[],[],[],[[]]]]

[.,[[.,[.,[.,[.,.]]]],.]] => [[],[[],[],[],[]]] => [[[[[[]]]],[]]]

[.,[[.,[.,[[.,.],.]]],.]] => [[],[[],[],[[]]]] => [[[[[],[]]],[]]]

[.,[[.,[[.,.],[.,.]]],.]] => [[],[[],[[]],[]]] => [[[[],[[]]],[]]]

[.,[[.,[[.,[.,.]],.]],.]] => [[],[[],[[],[]]]] => [[[[[]],[]],[]]]

[.,[[.,[[[.,.],.],.]],.]] => [[],[[],[[[]]]]] => [[[[],[],[]],[]]]

[.,[[[.,.],[.,[.,.]]],.]] => [[],[[[]],[],[]]] => [[[],[[[]]],[]]]

[.,[[[.,.],[[.,.],.]],.]] => [[],[[[]],[[]]]] => [[[],[[],[]],[]]]

[.,[[[.,[.,.]],[.,.]],.]] => [[],[[[],[]],[]]] => [[[[]],[[]],[]]]

[.,[[[[.,.],.],[.,.]],.]] => [[],[[[[]]],[]]] => [[[],[],[[]],[]]]

>>> Load all 2055 entries. <<<

Map

to ordered tree: right child = right brother

Description

Return an ordered tree of size

$n+1$ by the following recursive rule:

if $x$ is the right child of $y$ , $x$ becomes the right brother of $y$ ,
if $x$ is the left child of $y$ , $x$ becomes the first child of $y$ .

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.