Cardinal sequences and Cohen real extensions

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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 (2aleph₀)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.