Substitution Invariant Sturmian Words and Binary Trees

We take a global view at substitution invariant Sturmian sequences. We show that homogeneous substitution invariant Sturmian sequences sα,α can be indexed by two binary trees, associated directly with Johannes Kepler’s tree of harmonic fractions from 1619. We obtain similar results for the inhomogeneous sequences sα,1−α and sα,0