  We continue the study of prime simple modules for quantum affine algebras
from the perspective of $q$-fatorization graphs. In this paper we establish
several properties related to simple modules whose $q$-factorization graphs are
afforded by trees. The two most important of them are proved for type $A$. The
first completes the classification of the prime simple modules with three
$q$-factors by giving a precise criterion for the primality of a $3$-vertex
line which is not totally ordered. Using a very special case of this criterion,
we then show that a simple module whose $q$-factorization graph is afforded by
an arbitrary tree is real. Indeed, the proof of the latter works for all types,
provided the aforementioned special case is settled in general.
