A tall space with small bottom

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if kappa^{<kappa}=kappa then there is such a space of height kappa^+ with only kappa many isolated points. This implies that there is a locally compact scattered space of height omega_2 with omega_1 isolated points in ZFC, solving an old problem of the first author.