We study the simplex of probability measures on a Bratteli
diagram B
which are invariant with respect to the tail equivalence relation.
We prove a criterion of unique ergodicity of a Bratteli diagram.
In case when a finite rank $k$ Bratteli diagram B has $1 \leq l \leq k$
ergodic invariant measures,
we describe the structures of the diagram and the subdiagrams which
support these measures.
This is a joint work with S. Bezuglyi and J. Kwiatkowski.