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 $\frac{n}{m}$ can be described by the (unique part of) Ducci matrix sequence associated with $\frac{n}{m}$. 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.

已知通过反转 Stern-Brocot 树中的路径获得的树与 Calkin-Wilf 树重合 [J. Gibbons、D. Lester 和 R. Bird，FUNCTIONAL PEARL：枚举有理数，J. Funct。编程 16 (2006)，第 281-291 页]。受此启发，我们将介绍和研究通过反转 Ducci 树中的路径获得的树 R-DT。我们将证明树 R-DT 中从根到顶点的路径$\frac{n}{米}$ 可以通过（唯一的）杜奇矩阵序列来描述 $\frac{n}{米}$. 然后，我们查看 Stern-Brocot 树和树 R-DT 之间的关系，并提出这些树的每个级别之间的转换算法等。使用这些结果，我们将证明树 R-DT 的许多属性。