## Abstract

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set of labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the set of ternary trees, the set of quadrangular dissections, and the set of flagged Perfectly Chain Decomposed Ditrees, and we give topological and/or combinatorial interpretations of the structural bijections between them. In particular the bijection from the set of quadrangular dissections to the set of non-crossing trees seems to be new. Further we give explicit formulas for the number of self-dual labeled and unlabeled non-crossing trees and the set of quadrangular dissections up to rotations and up to rotations and reflections.

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## Notes

- 1.
See for example [16] for the basic definitions of Fuss–Catalan numbers, called

*generalized Catalan numbers*there. - 2.
All the relevant notions and terminology are reviewed in Sect. 2.5.

- 3.
For an explanation of the term

*mind-body*see Section 2.3 of [1]. - 4.
Thanks to the anonymous referee of a previous version for pointing this out.

- 5.
- 6.
Called “symmetric ternary trees” there.

- 7.
See for example [27, sections 5.3 and 11.1], or any “Discrete Mathematics” textbook.

- 8.
For general pegs these two arcs could be the same, but this cannot happen for nc-trees, except in the degenerate case of the tree with no edges.

- 9.
First observed by Dénes in [5].

- 10.
Alternatively we can obtain the peg as the total space of a branched covering of the disk, see [1, Section 4.4].

- 11.
Recall that we use left and right exponential notation for conjugation. Since transpositions are involutions the distinction is mute in the current context; however, it is useful in more general contexts.

- 12.
What we called

*c*in Remark 2.16. - 13.
Essentially due to Moszowski [20].

- 14.
This is different than the formula in [2]. The inconsistency is due to different conventions on how to multiply transpositions and how exactly the braid group acts. With our conventions Armstrong’s formula would give \(K^{-1}\) that corresponds to the action of \(\Delta _m^{-1}\). Of course, \(K^{-1}\) is also a complement in the lattice.

- 15.
Actually when \(\Gamma \) is a tree they are required!

- 16.
Directed Acyclic Graph. This observation is essentially due to [8]. The definition of medial digraphs was inspired in part from that paper.

- 17.
Recall that a ditree is a digraph whose underlying graph is a tree.

- 18.
That is a

*linear extension*of the poset whose Hasse diagram is the dag. - 19.
The first one may even be called pointless.

- 20.
Thanks to the anonymous referee that brought this work to my attention.

- 21.
The enumeration of oriented unlabeled trees is not explicitly stated there but a formula can be deduced from the calculations.

- 22.
It follows that so is the triple \(\left( \mathcal {N}_m, \nu _q,\mathrm {C}_{m+1} \right) \).

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The Computer Algebra Systems Sage [7] and GAP [11] were used extensively to confirm calculations and check conjectures at several stages of this project. I would like to thank Cormac O’Sullivan for valuable comments. Finally, I would also like to thank the referees of this and earlier versions of the paper for invaluable comments and for suggesting interesting connections with the literature.

Communicated by Matjaz Konvalinka.

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Apostolakis, N. Non-crossing Trees, Quadrangular Dissections, Ternary Trees, and Duality-Preserving Bijections.
*Ann. Comb.* **25, **345–392 (2021). https://doi.org/10.1007/s00026-021-00531-w

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