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Identifier

Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees

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click to show experimental identities (only identities of compositions of up to three maps are shown)

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.

References

[1] de Bruijn, N. G. and Morselt, B. J. M. "A note on plane trees." J. Combinatorial Theory 2 (1967), 27–34. MR0205872
[2] Donaghey, Robert. "Automorphisms on Catalan trees and bracketings." J. Combin. Theory Ser. B 29 (1980), no. 1, 75–90. MR0584162
[3] Shapiro, Louis W. "The cycle of six." Fibonacci Quart. 17 (1979), no. 3, 253–259. MR0549808

Properties

bijective, graded

Sage code

def mapping(T):
    return T.to_binary_tree_right_branch().to_ordered_tree_left_branch()

Weight

Created

Jul 01, 2024 at 02:15 by Sam Hopkins

Updated

Jul 01, 2024 at 02:15 by Sam Hopkins