In this note we present some results on the Calkin-Wilf tree of irreducible fractions, giving an insight on the duality relating it to the Stern-Brocot tree, and proving noncommutative versions of known results relating labels of the CalkinWilf trees to hyperbinary expansions of positive integers. The main tool is the Christoffel tree introduced in a paper by Berstel and de Luca.

We address the problem of computing the exact characteristics of any subsegment of a digital straight line with known characteristics. We present a new algorithm that solves this problem, whose correctness is proved. Its principle is to climb the SternBrocot tree of fraction in a bottom-up way. Its worst-time complexity is proportionnal to the difference of depth of the slope of the input line ...