Compact subsets of the Baire space

Let P be the natural forcing for producing a finite splitting tree using finite conditions. Namely, p ∈ P iff p ⊆ ω is a finite subtree and p ≤ q iff p ⊇ q is an end extension of q. End extension means if s ∈ p\q then s ⊇ t for some t ∈ q which is terminal in q, i.e., has no extensions in q. This order is countable and hence forcing equivalent to adding a single Cohen real. The union of P-generic filter is a tree T ⊆ ω which determines a compact set K ⊆ ω by K = [T ].