  Given a Bratteli diagram B, we study the set O(B) of all possible orderings w
on a Bratteli diagram B and its subset P(B) consisting of `perfect' orderings
that produce Bratteli-Vershik dynamical systems (Vershik maps). We give
necessary and sufficient conditions for w to be perfect. On the other hand, a
wide class of non-simple Bratteli diagrams that do not admit Vershik maps is
explicitly described. In the case of finite rank Bratteli diagrams, we show
that the existence of perfect orderings with a prescribed number of extreme
paths affects significantly the values of the entries of the incidence matrices
and the structure of the diagram B. Endowing the set O(B) with product measure,
we prove that there is some j such that almost all orderings on B have j
maximal and minimal paths, and that if j is strictly greater than the number of
minimal components that B has, then almost all orderings are imperfect.
