Theorem. Bédard's figure is generated by a finite automaton.
The language of reduced words (as well as that of ShortLex words) for a Coxeter group is
regular (Brink & Howlett).
ShortLex tree for affine A2
and its minimal version
The set of words containing one side of a braid relation
(a braid word) is regular.
The set of reduced words with a unique reduced expression is that
of reduced words containing no braid words (Tits).
[ unique reduced words ]
= [ reduced words ] AND NOT [ containing braid word ]