  We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is a
countable compact space. Namely ${Dz}(C([0,\omega^{\omega^\alpha}])) =
\omega^{1+\alpha+1}$, whenever $0\le\alpha<\omega_1$. More generally,
${Dz}(C(K))=\omega^{1+\alpha+1}$ if $K$ is a scattered compact whose height
$\eta(K)$ satisfies $\omega^\alpha<\eta(K)\leq \omega^{\alpha+1}$ with an
$\alpha$ countable.
