  We study non-ascending HNN extensions acting on their Bass-Serre trees, and
characterize C*-simplicity and the unique trace property by means of the kernel
and quasi-kernels of the HNN extension in question. We also present a concrete
example of an HNN extension that is a new example of a group that is not
C*-simple but does have the unique trace property. Additionally, we include
certain more general results, mostly based on previous work of various authors,
concerning C*-simplicity of groups admitting extreme boundary actions, and in
particular, groups acting on trees.
