  A stable set in a graph G is a set of mutually non-adjacent vertices,
alpha(G) is the size of a maximum stable set of G, and core(G) is the
intersection of all its maximum stable sets. In this paper we demonstrate that
in a tree T, of order n greater than 1, any stable set of size greater or equal
to n/2 contains at least one pendant vertex. Hence, we deduce that any maximum
stable set in a tree contains at least one pendant vertex. Our main finding is
the theorem claiming that if T does not own a perfect matching, then at least
two pendant vertices an even distance apart belong to core(T). While it is
proved by Levit and Mandrescu that if G is a connected bipartite graph of order
at least 2, then the size of core(G) is different from 1, our new statement
reveals an additional structure of the intersection of all maximum stable sets
of a tree. The above assertions give refining of one result of Hammer, Hansen
and Simeone, stating that if a graph G is of order less than 2*alpha(G), then
core(G) is non-empty, and also of a result of Jamison, Gunter, Hartnel and
Rall, and Zito, saying that for a tree T of order at least two, the size of
core(G) is different from 1.
