We compare two sets of (infinite) binary sequences whose suffixes satisfy
extremal conditions: one occurs when studying iterations of unimodal
continuous maps from the unit interval into itself, but it also characterizes
univoque real numbers; the other is a disguised version of the set of
characteristic Sturmian sequences. As a corollary to our study we obtain
that a real number β in (1,2) is univoque and self-Sturmian if and
only if the β-expansion of 1 is of the form 1v, where v is a
characteristic Sturmian sequence beginning itself in 1.
