A tall space with a 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.