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 $M$?
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 $\ell$.