Sturmian words and the Stern sequence

Central, standard, and Christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart of characteristic Sturmian words. A natural arithmetization of the theory is obtained by representing central and Christoffel words by irreducible fractions labeling respectively two binary trees, the Raney (or Calkin-Wilf) tree and the Stern-Brocot tree. The sequence of denominators of the fractions in Raney's tree is the famous Stern di-atomic numerical sequence. An interpretation of the terms s(n) of Stern's sequence as lengths of Christoffel words when n is odd, and as minimal periods of central words when n is even, allows one to interpret several results on Christoffel and central words in terms of Stern's sequence and, conversely, to obtain a new insight in the combi-natorics of Christoffel and central words by using properties of Stern's sequence. One of our main results is a non-commutative version of the " alternating bit sets theorem " by Calkin and Wilf. We also study the length distribution of Christoffel words corresponding to nodes of equal height in the tree, obtaining some interesting bounds and inequalities.