DeBruijnMorselt from OrderedTree

The right-branch bijection to binary trees followed by the left-branch bijection back: the de Bruijn-Morselt automorphism of the ordered trees.

OrderedTree → OrderedTreebijection

FindStat's Mp00328. Chaotic in general: Shapiro shows its sixth power is the identity on the trees that correspond to compositions.

Called as CombinatorialMap(x, "DeBruijnMorselt"). The head's page has its examples.

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Pick a statistic on OrderedTree. Its distribution over the images of OrderedTree at each size is computed, and compared with the 21 statistics on OrderedTree: those that share it are the ones the map sends it to, as far as the sizes computed show.

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FindStat Mp00328

DeBruijn-Morselt plane tree automorphism

Matched by value: our map agrees with FindStat's image on each of the 197 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

BijectionGraded mapas FindStat records them, checked by FindStat's contributors.

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 correspo…

Object	Image
`[]`	`[]`
`[[]]`	`[[]]`
`[[[]]]`	`[[],[]]`
`[[],[]]`	`[[[]]]`

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00328 has the rest.