  The concept of pseudo q-factorization graphs was recently introduced by the
last two authors as a combinatorial language which is suited for capturing
certain properties of Drinfeld polynomials. Using certain known representation
theoretic facts about tensor products of Kirillov Reshetikhin modules and
qcharacters, combined with special topological/combinatorial properties of the
underlying q-factorization graphs, the last two authors showed that, for
algebras of type A, modules associated to totally ordered graphs are prime,
while those associated to trees are real. In this paper, we extend the latter
result. We introduce the notions of strongly real modules and that of trees of
modules satisfying certain properties. In particular, we can consider snake
trees, i.e., trees formed from snake modules. Among other results, we show that
a certain class of such generalized trees, which properly contains the snake
trees, give rise to strongly real modules.
