Donaghey's transformation: carousel effects and tame components

In this paper, Donaghey's transformation is investigated. It is a combinatorial dynamical system, based on the combinatorial interpretations of Catalan numbers, with the transition function of it being the composition of the direct and the inverse bijections between cubic and non-cubic trees. The dynamical system under investigation operates on a finite set and is inversible; therefore it is a mere permutation of trees. The properties of the cycles of this permutation, called the orbits, are studied in terms of permutation of structural elements of trees. In the course of studies, the systematic initiation of particular effects is indicated. These particular effects are referred to as “carousel”: it is the movement of objects from one classification category to another, typical of natural classifications. In this form, they look as an indicator of system complexity. Two new carousel effects for structural elements, referred to as triads and germs, are described. The carousel effect for triads is used for the detection of two families of trees; the lengths of all tree arcs in the first family are equal to one; in the second family, they are equal to two. Here, the term “the arcs of the orbit” is used to denote its fragments between two trees, which have no left subtrees. Therefore, the properties of the arcs are the global properties of the orbits, and the carousel effects are local. The corresponding enumeration problems are solved: it is demonstrated that the number of trees of the first family increases as $\dfrac{C}{n^{3/2}}\left(\dfrac{5}{2^{4/3}-2^{2/3}+1}\right)^n$, the number of trees of the second family — as $\dfrac{3^{n+1/2}}{n^{3/2}\sqrt{\pi}}$ ($n$ is the number of triads). The paper presents the family of cycles with the length 6, which are different from the ones discovered by L. Shapiro, the number of which increases as $\Theta(n^2)$, and the family of cycles with length 9, the number of which increases as $\Theta(2^{n/2})$. A class of orbits with the lengths growing up as $\Theta(n^2)$ is detected.