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We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is acountable 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.

Weak$^*$ dentability index of spaces $C([0,\alpha])$