On almost disjoint families and Johnson-Lindenstrauss spaces

Every almost disjoint family $A$ of infinite subsets of $\omega$ gives rise to a scattered compact space $K_A$ and a family of Johnson-Lindenstrauss spaces JL${}_p(A)$. We consider almost disjoint families arising from finitely branching trees and investigate whether they define homeomorphic compacta and isomorphic Banach spaces.