  In this article we determine the Brauer trees of the unipotent blocks with
cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. The
degrees of the unipotent characters of these objects were given by Lusztig, and
using the general theory of perverse equivalences we can reconstruct the Brauer
trees that would be consistent with Deligne--Lusztig theory and the geometric
version of Brou\'e's conjecture. We construct the trees using standard
arguments whenever possible, and check that the Brauer trees predicted by
Brou\'e's conjecture are consistent with both the mathematics and philosophy of
blocks with cyclic defect groups.

Representation Theory