We continue our discussion of perfect and scattered subsets in the generalized Cantor space. This week we finish the proof of the fact that \kappa-closed forcings don't add branches to \kappa-scattered subsets of 2^{\kappa}. We then introduce a collection of topologies over 2^{\lambda} whose restrictions to P_{\kappa}\lambda have some desirable properties. These topologies will rely on the notion of a P_{\kappa}\lambda-forest, which is a natural generalization of a tree.