  We characterize the weak-star point of continuity property for subspaces of
dual spaces with separable predual and we deduce that the weak-star point of
continuity property is determined by subspaces with a Schauder basis in the
natural setting of dual spaces of separable Banach spaces. As a consequence of
the above characterization we get that a dual space satisfies the Radon-Nikodym
property if, and only if, every seminormalized topologically weak-star null
tree has a boundedly complete branch, which improves some results in \cite{DF}
obtained for the separable case. Also, as a consequence of the above
characterization, the following result obtained in \cite{R1} is deduced: {\it
every seminormalized basic sequence in a Banach space with the point of
continuity property has a boundedly complete subsequence
