Suppose we are given a rooted tree $T$, and a set of vertices $M$ that separates the root of $T$ from its leaves. In other words, every path from the root of $T$ to a leaf contains a vertex in $M$. Is there a standard term for such a set $M$?

$\begingroup$ If you want to keep the arborial metaphor going, perhaps "complete pruning" would be appropriate. $\endgroup$ – Ryan Budney Mar 17 at 2:14

$\begingroup$ If this doesn't have a name yet, I recommend calling it a topping. $\endgroup$ – Jesko Hüttenhain Mar 17 at 2:21

$\begingroup$ How about "nonleaves"? $\endgroup$ – David G. Stork Mar 21 at 0:25

$\begingroup$ @DavidG.Stork The set $M$ may contain leaves. For example, note that if we choose $M$ to be the set of all leaves, then it indeed satisfies the property that every path from the root to a leaf passes through $M$. $\endgroup$ – Or Meir Mar 21 at 13:33

$\begingroup$ Define "through." $\endgroup$ – David G. Stork Mar 21 at 13:46
Call the root node $r$. If you join each leaf node to a dummy sink node $t$, then $M$ would be an $(r,t)$ vertex separator, also known as vertex cut or separating set.
Even without the dummy node, $M$ is an $(r,\ell)$ vertex separator for each leaf node $\ell$.