Laver forcing, trees and stems

Question

In the following snippet about Laver forcing I do not understand several things:

what is the intuition behind that $T$ extends $S$ if $T\subseteq S$. Second, is by a stem meant ambiguously any node of the linearly ordered beginning of the tree in question, from the first to last such node, or only unique last such node ? So, are there many stems or just one ? Or is by stem even meant all of them together ? In the -5th paragraph they say the stem. Finally what is $\prec$ in -6th paragraph, is it any relation satisfying the subsequent condition ?

Answer 1

Second, is by a stem meant ambiguously any node of the linearly ordered beginning of the tree in question, from the first to last such node, or only unique last such node ?

The stem is the unique last node (this is implied by the second bulletpoint, which says that every node not below the stem has infinitely many successors - including the stem itself). So every condition has a unique stem.

what is the intuition behind that $T$ extends $S$ if $T\subseteq S$?

We usually think of conditions as building the generic, but we can also think of them as constraining the generic: stronger conditions put stricter restrictions on what the generic can possibly be, with the generic filter itself ultimately putting a complete set of constraints. This all means that more constraining conditions should be stronger. Since the way a Laver condition constrains the generic is by limiting its possible initial segments to those which are nodes of the tree, smaller trees yield stricter restrictions.

Finally what is $\prec$ in $6$th paragraph, is it any relation satisfying the subsequent condition ?

Yes.