  We study which $\kappa$-distributive forcing notions of size $\kappa$ can be
embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters
under various large cardinal assumptions. An alternative formulation -- can the
filter of dense open subsets of a $\kappa$-distributive forcing notion of size
$\kappa$ be extended to a $\kappa$-complete ultrafilter.
