  We classify t-structures and thick subcategories in discrete cluster
categories $\mathcal{C}(\mathcal{Z})$ of Dynkin type $A$, and show that the set
of all t-structures on $\mathcal{C}(\mathcal{Z})$ is a lattice under inclusion
of aisles, with meet given by their intersection. We show that both the lattice
of t-structures on $\mathcal{C}(\mathcal{Z})$ obtained in this way and the
lattice of thick subcategories of $\mathcal{C}(\mathcal{Z})$ are intimately
related to the lattice of non-crossing partitions of type $A$. In particular,
the lattice of equivalence classes of non-degenerate t-structures on such a
category is isomorphic to the lattice of non-crossing partitions of a finite
linearly ordered set.
