  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.
