  For $r \geq 1$ and $\mathbf{n} \in
\mathbb{Z}_{\geq0}^r\setminus\{\mathbf{0}\}$, we construct the compactified
moduli space $\overline{2\mathcal{M}}_{\mathbf{n}}$ of witch curves of type
$\mathbf{n}$. We equip $\overline{2\mathcal{M}}_{\mathbf{n}}$ with a
stratification by the 2-associahedron $W_{\mathbf{n}}$, and prove that
$\overline{2\mathcal{M}}_{\mathbf{n}}$ is compact and metrizable. In addition,
we show that the forgetful map $\overline{2\mathcal{M}}_{\mathbf{n}} \to
\overline{\mathcal{M}}_r$ to the moduli space of stable disk trees is
continuous and respects the stratifications.
