We study the topology of
codimension one taut foliations of closed orientable 3-manifolds which are smooth
along the leaves. In particular, we focus on the lifts of these foliations to the universal
cover, specifically when any set of leaves corresponding to nonseparable points in the
leaf space can be totally ordered. We use the structure of branching in the
lifted foliation to find conditions that ensure two nonseparable leaves are
left invariant under the same covering translation. We also determine when
the set of leaves nonseparable from a given leaf is finite up to the action of
covering translations. The hypotheses for the results are satisfied by all Anosov
foliations.
