Non-crossing Trees, Quadrangular Dissections, Ternary Trees, and Duality-Preserving Bijections

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