Generating Stern-Brocot Type Rational Numbers with Mediants

The Stern–Brocot tree is a method of generating or organizing all fractions in the interval \((0,1)\) by starting with the endpoints \(\frac{0}{1} \) and \(\frac{1}{1}\) and repeatedly applying the mediant operation: \(m\left( \frac{a}{b},\frac{c}{d} \right) =\frac{a+c}{b+d}\). A recent paper of Aiylam considers two generalizations: one is to apply the mediant operation starting with an arbitrary interval \(\left( \frac{a}{b},\frac{c}{d} \right)\) (the fractions must be non-negative), and the other is to allow arbitrary reduction of generated fractions to lower terms. In the present paper, we give simpler proofs of some of Aiylam's results, and we give a simpler method of generating just the portion of the tree that leads to a given fraction.