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

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.