  Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit
cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the
existence of a normal $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here,
it is proved that the same conclusion remains valid after replacing the
hypothesis $\square^*_\lambda$ by $\square(\lambda^+,{<}\lambda)$.
  As $\square(\lambda^+,{<}\lambda)$ does not impose a bound on the order-type
of the witnessing clubs, our construction is necessarily different from that of
Ben-David and Shelah, and instead uses walks on ordinals augmented with club
guessing.
  A major component of this work is the study of postprocessing functions and
their effect on square sequences. A byproduct of this study is the finding that
for $\kappa$ regular uncountable, $\square(\kappa)$ entails the existence of a
partition of $\kappa$ into $\kappa$ many fat sets. When contrasted with a
classic model of Magidor, this shows that it is equiconsistent with the
existence of a weakly compact cardinal that $\omega_2$ cannot be split into two
fat sets.
