In "A proof of Vaught's conjecture for ω-stable theories," the
notions of ENI-NDOP and eni-depth have been introduced, which are variants of
the notions of NDOP and depth known from Shelah's classification theory. First,
we show that for an ω-stable first-order complete theory, ENI-NDOP
allows tree decompositions of countable models. Then we discuss the
relationship between eni-depth and the complexity of the isomorphism relation
for countable models of such a theory in terms of Borel reducibility as
introduced by Friedman and Stanley and construct, in particular, a sequence of
complete first-order ω-stable theories $(T_\alpha)_{\alpha < \omega_1}$ with increasing and cofinal eni-depth and isomorphism relations which
are strictly increasing with respect to Borel reducibility.