Geodesic
Donaghey's transformation: an elementary approach
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 148-177

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In this paper, we study the properties of the orbit of a special transformation of plane planted trees and its remarkable behavior. This transformation was introduced by R. Donaghey. We prove some nontrivial properties of this transformation (the behavior of the transformation of the fringed trees, “carousel effect”) and obtain a lower estimate of the number of orbits of the transformation. 
@article{ZNSL_2013_411_a9,
     author = {I. A. Pushkarev and V. A. Byzov},
     title = {Donaghey's transformation: an elementary approach},
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     pages = {148--177},
     publisher = {mathdoc},
     volume = {411},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a9/}
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I. A. Pushkarev; V. A. Byzov. Donaghey's transformation: an elementary approach. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 148-177. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a9/