Author: Kos Diveris email: diveris@stolaf.edu Address: Department of Mathematics, St. Olaf College, Northfield, MN 55057, USA Author: Marju Purin email: purin@stolaf.edu Address: Department of Mathematics, St. Olaf College, Northfield, MN 55057, USA Author: Peter Webb email: webb@math.umn.edu Address: School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Title: Combinatorial restrictions on the tree class of the Auslander-Reiten quiver of a triangulated category Abstract: We show that if a connected, Hom-finite, Krull-Schmidt triangulated category has an Auslander-Reiten quiver component with Dynkin tree class then the category has Auslander-Reiten triangles and that component is the entire quiver. This is an analogue for triangulated categories of a theorem of Auslander, and extends a previous result of Scherotzke. We also show that if there is a quiver component with extended Dynkin tree class, then other components must also have extended Dynkin class or one of a small set of infinite trees, provided there is a non-zero homomorphism between the components. The proofs use the theory of additive functions.