A calculational approach to path-based properties of the Eisenstein-Stern and Stern-Brocot trees via matrix algebra

Abstract

This paper proposes a calculational approach to prove properties of two well-known binary trees used to enumerate the rational numbers: the Stern-Brocot tree and the Eisenstein-Stern tree (also known as Calkin-Wilf tree). The calculational style of reasoning is enabled by a matrix formulation that is well-suited to naturally formulate path-based properties, since it provides a natural way to refer to paths in the trees. Three new properties are presented. First, we show that nodes with palindromic paths contain the same rational in both the Stern-Brocot and Eisenstein-Stern trees. Second, we show how certain numerators and denominators in these trees can be written as the sum of two squares $x^2$ and $y^2$, with the rational $x/y$ appearing in specific paths. Finally, we show how we can construct Sierpiński’s triangle from these trees of rationals.

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