Continued fraction algorithm for Sturmian colorings of trees

Factor complexity $b_\phi(n)$ for a vertex coloring $\phi$ of a regular tree is the number of colored $n$-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity $b_\phi(n) = n+2$. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to induction algorithm of Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data for the induction algorithm.