  In this paper we focus on Bratteli-Vershik models of general compact
zero-dimensional systems with the action of a homeomorphism. An ordered
Bratteli diagram is called decisive if the corresponding Vershik map prolongs
in a unique way to a homeomorphism of the whole path space of the Bratteli
diagram. We prove that a compact invertible zero-dimensional system has a
decisive Bratteli-Vershik model if and only if the set of aperiodic points is
either dense, or its closure misses one periodic orbit.
