  For every well founded tree $\mathcal{T}$ having a unique root such that
every non-maximal node of it has countable infinitely many immediate
successors, we construct a $\mathcal{L}_\infty$-space $X_{\mathcal{T}}$. We
prove that for each such tree $\mathcal{T}$, the Calkin algebra of
$X_{\mathcal{T}}$ is homomorphic to $C(\mathcal{T})$, the algebra of continuous
functions defined on $\mathcal{T}$, equipped with the usual topology. We use
this fact to conclude that for every countable compact metric space $K$ there
exists a $\mathcal{L}_\infty$-space whose Calkin algebra is isomorphic, as a
Banach algebra, to $C(K)$.
