  For a Bratteli diagram $B$, we study the simplex $\mathcal{M}_1(B)$ of
probability measures on the path space of $B$ which are invariant with respect
to the tail equivalence relation. Equivalently, $\mathcal{M}_1(B)$ is formed by
probability measures invariant with respect to a homeomorphism of a Cantor set.
We study relations between the number of ergodic measures from
$\mathcal{M}_1(B)$ and the structure and properties of the diagram $B$. We
prove a criterion and find sufficient conditions of unique ergodicity of a
Bratteli diagram, in which case the simplex $\mathcal{M}_1(B)$ is a singleton.
For a finite rank $k$ Bratteli diagram $B$ having exactly $l \leq k$ ergodic
invariant measures, we explicitly describe the structure of the diagram and
find the subdiagrams which support these measures. We find sufficient
conditions under which: (i) a Bratteli diagram has a prescribed number (finite
or infinite) of ergodic invariant measures, and (ii) the extension of a measure
from a uniquely ergodic subdiagram gives a finite ergodic invariant measure.
Several examples, including stationary Bratteli diagrams, Pascal-Bratteli
diagrams, and Toeplitz flows, are considered.
