- FindStat

Identifier

Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees

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Map

to ordered tree: left child = left brother

Description

Return an ordered tree of size

$n+1$ by the following recursive rule:

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