Counting spaces of functions on separable compact lines

We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the supremum norm: Let $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ are there, for $K\in \mathcal{K}$? We prove that for any uncountable regular cardinal number $κ$, there exist exactly $2^κ$ isomorphism types of spaces $C(K)$ for compact spaces of weight $κ$. We show that, for the class $\mathcal{L}_{ω_1}$ of separable compact linearly ordered spaces of weight $ω_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{ω_1}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{ω_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type.