Reversing the paths in the Ducci tree

Abstract

It is known that the tree obtained by reversing the paths in the Stern–Brocot tree coincides with the Calkin–Wilf tree [J. Gibbons, D. Lester, and R. Bird, FUNCTIONAL PEARL: Enumerating the rationals, J. Funct. Programming 16 (2006), pp. 281–291]. Motivated by this fact, we shall introduce and study the tree R-DT obtained by reversing the paths in the Ducci tree. We shall show that the path in the tree R-DT from the root to a vertex can be described by the (unique part of) Ducci matrix sequence associated with . We then look at the relationships between the Stern–Brocot tree and the tree R-DT and present, among other things, algorithms for conversion between each levels of these trees. Using these results, we shall prove a number of properties of the tree R-DT.