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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 $\lambda$. Theyshowed that if $\kappa$ is singular of countable cofinality, then$\mathbb{P}_\kappa$ is minimal for $\omega$-sequences assuming that $\kappa$ isa supremum of a sequence of measurable cardinals. We obtain this result withoutthe measurability assumption.  Prikry proved that $\mathbb{P}_\kappa$ is $(\omega,\nu)$-distributive for all$\nu<\kappa$ given a singular $\omega$-strong limit cardinal $\kappa$ ofcountable cofinality, and Dobrinen et al$.$ asked whether this resultgeneralizes if $\kappa$ has uncountable cofinality. We answer their question inthe negative by showing that $\mathbb{P}_\kappa$ is not$(\lambda,2)$-distributive if $\kappa$ is a $\lambda$-strong limit ofuncountable cofinality $\lambda$ and we obtain the same result for a range ofsimilar forcings, including one that Dobrinen et al$.$ consider that consistsof pre-perfect trees. We also show that $\mathbb{P}_\kappa$ in particular isnot $(\omega,\cdot,\lambda^+)$-distributive under these assumptions.  While developing these ideas, we address natural questions regardingminimality and collapses of cardinals.

Distributivity and Minimality in Perfect Tree Forcings for Singular Cardinals