Given a tree $T\subseteq \omega^{<\omega}$ we call T superperfect if
- For all $t \in T$ there is an $s\supseteq t$ such that $s^\smallfrown n \in T$ for all $n\in\omega$, where $s^\smallfrown n$ is the finite sequence in which we add $n$ at the end of $s$.
Has the forcing notion $(P,\supseteq)$ of superperfect trees (in the above sense) been studied ? Is it known under a different name?
This forcing is introduced and studied, among other things, in:
Newelski, L.; Rosłanowski, A., The ideal determined by the unsymmetric game, Proc. Am. Math. Soc. 117, No. 3, 823-831 (1993). ZBL0778.03016.
They denote this forcing by $\mathbb{D}_\omega$. They show (Theorem 2.1) that this forcing adds a Cohen real and that it doesn't add any random or dominating reals. I don't know if it has been studied further in more recent papers.