Combinatorial restrictions on the tree class of the Auslander–Reiten quiver of a triangulated category

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.