The topological types of hyperspaces of 0-dimensional compacta

Let X be a 0-dimensional compactum (= compact metric space) and let 2X denote the hyperspace of X, the set of non-empty closed subsets of X with the Hausdorff metric. We shall determine all the topological types of the hyperspaces 2X. They are only finitely many in the non-trivial case where the scattered part of X contains infinitely many points.