  We introduce notions of stationarily ordered types and theories; the latter
generalizes weak o-minimality and the first is a relaxed version of weak
o-minimality localized at the locus of a single type. We show that forking, as
a binary relation on elements realizing stationarily ordered types, is an
equivalence relation and that each stationarily ordered type in a model
determines some order-type as an invariant of the model. We study weak and
forking non-orthogonality of stationarily ordered types, show that they are
equivalence relations and prove that invariants of non-orthogonal types are
closely related. The developed techniques are applied to prove that in the case
of a binary, stationarily ordered theory with fewer than $2^{\aleph_0}$
countable models, the isomorphism type of a countable model is determined by a
certain sequence of invariants of the model. In particular, we confirm Vaught's
conjecture for binary, stationarily ordered theories.
