Selective Independence and $h$-Perfect Tree Forcing Notions

Generalizing the proof for Sacks forcing, we show that the $h$-perfect tree forcing notions introduced by Goldstern, Judah and Shelah preserve selective independent families even when iterated. As a result we obtain new proofs of the consistency of $\mathfrak{i} = \mathfrak{u} < $\mathrm{n}\mathrm{o}\mathrm{n}$ (\mathcal N) = $\mathrm{c}\mathrm{o}\mathrm{f}$ (\mathcal N)$and$\mathfrak{i} < \mathfrak{u} = $\mathrm{n}\mathrm{o}\mathrm{n}$ (\mathcal N) = $\mathrm{c}\mathrm{o}\mathrm{f}$( \mathcal N)$ as well as some related results.