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.

使用在较早的工作中开发的适当嵌入图论，我们在标记的非穿越树的集合上定义了对合对偶性，从而提升了在未标记的非穿越树的集合中的明显对偶性。非相交树的集合是具有一个生成器的自由三元岩浆，并且该对偶性是在任何此类岩浆中定义的对偶性的一个实例。具有一个生成器的任何两个自由三元岩浆都是通过一个独特的同构而同构的，我们称其为结构双射。除了非交叉树集之外，我们还将三元树集，四边形剖分集和标记的完全链分解二叉树集作为一个生成器，作为自由三元岩浆，并且对以下内容进行拓扑和/或组合解释：它们之间的结构双射。特别是从四边形剖分集到非交叉树集的双射似乎是新的。此外，我们为自对偶标记和未标记的非交叉树的数目以及直至旋转以及旋转和反射的四边形剖分集提供了明确的公式。