  Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_\kappa$
consisting of perfect trees of height $\lambda$ and width $\kappa$ where
$\kappa$ is a singular $\omega$-strong limit of cofinality $\lambda$. They
showed that if $\kappa$ is singular of countable cofinality, then
$\mathbb{P}_\kappa$ is minimal for $\omega$-sequences assuming that $\kappa$ is
a supremum of a sequence of measurable cardinals. We obtain this result without
the measurability assumption.
  Prikry proved that $\mathbb{P}_\kappa$ is $(\omega,\nu)$-distributive for all
$\nu<\kappa$ given a singular $\omega$-strong limit cardinal $\kappa$ of
countable cofinality, and Dobrinen et al$.$ asked whether this result
generalizes if $\kappa$ has uncountable cofinality. We answer their question in
the negative by showing that $\mathbb{P}_\kappa$ is not
$(\lambda,2)$-distributive if $\kappa$ is a $\lambda$-strong limit of
uncountable cofinality $\lambda$ and we obtain the same result for a range of
similar forcings, including one that Dobrinen et al$.$ consider that consists
of pre-perfect trees. We also show that $\mathbb{P}_\kappa$ in particular is
not $(\omega,\cdot,\lambda^+)$-distributive under these assumptions.
  While developing these ideas, we address natural questions regarding
minimality and collapses of cardinals.
