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

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Highlights

  • This paper proposes a calculational approach to prove properties of the Eisenstein–Stern and the Stern–Brocot trees.

  • The calculational style of reasoning is enabled by a matrix formulation well-suited to formulate path-based properties.

  • We show that nodes with palindromic paths contain the same rational in both the Eisenstein–Stern and Stern–Brocot trees.

  • We show how certain numerators and denominators in these trees can be written as the sum of two squares.

  • We show how we can construct Sierpinski's triangle from these trees of rationals.

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 x2 and y2, with the rational xy appearing in specific paths. Finally, we show how we can construct Sierpiński's triangle from these trees of rationals.

Keywords

Stern–Brocot tree
Eisenstein–Stern tree (aka Calkin–Wilf tree)
Calculational method
Euclid's algorithm
Sierpiński's triangle
Rational number

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