How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$?

Question

To make the question more precise:

We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$.

Let $\mathcal{C}$ be a collection of compact and zero-dimensional topologies on $\mathbb{N}$ such that for $\tau_1 \neq \tau_2 \in \mathcal{C}$ there is no homeomorphism $\varphi: (\mathbb{N},\tau_1) \to (\mathbb{N},\tau_2)$.

Question: What is the maximal cardinality that $\mathcal{C}$ can have?

Answer 1

By a theorem of Mazurkiewicz and Sierpinski, every countable compact Hausdorff space is homeomorphic to an ordinal, so there are at most $\aleph_1$ of them. Conversely, any countable ordinal can be the Cantor-Bendixson rank of a countable compact Hausdorff space (easy to prove by induction), so there are indeed $\aleph_1$ different countable compact Hausdorff spaces.