  In this article, we continue the study of a certain family of 2-Calabi-Yau
tilted algebras, called dimer tree algebras. The terminology comes from the
fact that these algebras can also be realized as quotients of dimer algebras on
a disc. They are defined by a quiver with potential whose dual graph is a tree,
and they are generally of wild representation type. Given such an algebra $B$,
we construct a polygon $\mathcal{S}$ with a checkerboard pattern in its
interior, that defines a category $\text{Diag}(\mathcal{S})$. The
indecomposable objects of $\text{Diag}(\mathcal{S})$ are the 2-diagonals in
$\mathcal{S}$, and its morphisms are certain pivoting moves between the
2-diagonals. We prove that the category $\text{Diag}(\mathcal{S})$ is
equivalent to the stable syzygy category of the algebra $B$. This result was
conjectured by the authors in an earlier paper, where it was proved in the
special case where every chordless cycle is of length three.
  As a consequence, we conclude that the number of indecomposable syzygies is
finite, and moreover the syzygy category is equivalent to the 2-cluster
category of type $\mathbb{A}$. In addition, we obtain an explicit description
of the projective resolutions, which are periodic. Finally, the number of
vertices of the polygon $\mathcal{S}$ is a derived invariant and a singular
invariant for dimer tree algebras, which can be easily computed form the
quiver.
