Ramsey families of subtrees of the dyadic tree

We show that for every rooted, finitely branching, pruned tree $T$of height $\omega$ there exists a family $\mathcal{F}$ which consists of order isomorphic to $T$ subtrees of the dyadic tree $C=\{0,1\}^{<\mathbb{N} }$ with the following properties: (i) the family $\mathcal{F}$ is a $G_\delta$ subset of $2^C$; (ii) every perfect subtree of $C$ contains a member of $\mathcal{F}$; (iii) if $K$ is an analytic subset of $\mathcal{F}$, then for every perfect subtree $S$ of $C$ there exists a perfect subtree $S'$ of $S$ such that the set $\{A\in\mathcal{F}: A\subseteq S'\}$ either is contained in or is disjoint from $K$.