  We show that if we add any number of Cohen reals to the ground model then, in
the generic extension, a locally compact scattered space has at most
(2^{aleph_0})^V many levels of size omega. We also give a complete ZFC
characterization of the cardinal sequences of regular scattered spaces.
Although the classes of the regular and of the 0-dimensional scattered spaces
are different, we prove that they have the same cardinal sequences.
