Equivalents of NOTOP

Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type $V$-dominated by an independent triple is isolated over the triple. If $T$ has NOTOP, then every model $N$ is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah’s assertion from Chapter XII of \cite{Shc} that NOTOP implies PMOP (without using NDOP).