I don't know if it will help you, but there is another reformulation that may be useful here.

You can model the infinite complete binary tree $\{0,1\}^{\omega}$ by the interval $[0,1]$ (with the binary expansion). In particular, if you fix a finite binary string $x$, then the subtree consisting of the branch $x$ followed by a complete binary tree represents a subinterval of $[0,1]$ of length $2^{-|x|}$.

So finding a subtree satisfying your condition 1 amounts to finding a ~~subset of $[0,1]$ with (Lebesgue) measure zero~~ closed subset with empty interior (thanks Blass).

The condition 2 is related to Hausdorff dimension of subsets of $[0,1]$. Indeed, the collection of vertices at distance $n$ from the root can be seen as a collection of subintervals, each of length $2^{-n}$, whose union covers your subset. The number of vertices is then the number of such intervals.

Roughly speaking, if you take a subset of dimension $\theta$ (Hausdorff or a variant, I don't remember exactly), then the number of vertices of your subtree at distance $n$ grows like $O(\theta^n)$ when $n \rightarrow \infty$.

pb