  One theorem of Nemhauser and Trotter ensures that, under certain conditions,
a stable set of a graph G can be enlarged to a maximum stable set of this
graph. For example, any stable set consisting of only simplicial vertices is
contained in a maximum stable set of G. In this paper we demonstrate that an
inverse assertion is true for trees of order greater than one, where, in fact,
all the simplicial vertices are pendant. Namely, we show that any maximum
stable set of such a tree contains at least one pendant vertex. Moreover, we
prove that if T does not own a perfect matching, then a stable set, consisting
of at least two pendant vertices, is included in the intersection of all its
maximum stable sets. For trees, the above assertion is also a strengthening of
one result of Hammer, Hansen, and Simeone, stating that if half of order of G
is less than the cardinality of a maximum stable set of G, then the
intersection of all its maximum stable sets is non-empty.
