  We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary
and directed, and whose trees are infinite, but for which the subtrees attached
to each vertex are as short as possible. For $d\geq3$, two independent copies
of such forests, pointing in opposite directions, can be pruned so as to become
disjoint. From this, we construct in $d\geq3$ a stationary, polynomially mixing
and uniformly elliptic environment of nearest-neighbor transition probabilities
on $\mathbb{Z}^d$, for which the corresponding random walk disobeys a certain
zero--one law for directional transience.
