  Consider the solutions $u$ to the elliptic equation $\mathcal{L}(u) =
\partial_i(a^{ij}(x) \partial_j u) + b^i(x) \partial_i u + c(x) u= 0$ with
$a^{ij}$ assumed only to be H\"older continuous. In this paper we prove an
explicit bound for $(n-2)$-dimensional Minkowski estimates of singular set
$\mathcal{S}(u) = \{ x \in B_1 : u(x) = |\nabla u(x)| = 0\}$ and critical set
$\mathcal{C}(u) \equiv \{ x\in B_{1} : |\nabla u(x)| = 0 \}$ in terms of the
bound on doubling index, depending on $c \equiv 0$ or not. Here the H\"older
assumption is sharp as it is the weakest condition in order to define the
critical set of $u$ according to elliptic estimates. We can also obtain an
optimal improvement on Cheeger-Naber-Valtorta's volume estimates on each
quantitative stratum $\mathcal{S}^k_{\eta, r}$. The main difficulty in this
situation is the lack of monotonicity formula which is essential to the
quantitative stratification. In our proof, one key ingredient is a new almost
monotonicity formula for doubling index under the H\"older assumption. Another
key ingredient is the quantitative uniqueness of tangent maps. It deserves to
note that our almost monotonicity is sufficient to address all the difficulties
arising from the absence of monotonicity in the analysis of differential
equations. We believe the idea could be applied to other relevant study.
