  A family of sets is intersecting if every pair of its sets intersect. A star
is a family with some element (a center) in each of its sets. The classical
1961 result of Erd\H{o}s, Ko, and Rado states that every intersecting family of
r-sets with $r\leq n/2$ has size at most that of a star. We say that graph G is
r-EKR if, among all intersecting families of independent r-sets of G, the
largest is attained by a star. In 2005 Holroyd and Talbot conjectured that
every graph G is r-EKR for all $1\leq r\leq \mu(G)/2$, where $\mu(G)$ is the
size of the smallest maximal independent set in G. We verified the conjecture
in 2011 for all chordal graphs containing an isolated vertex.
  For graphs without isolated vertices it is difficult to determine the center
of the largest star, which is often necessary to prove that they are EKR. A
tree has the leaf property if its largest star occurs on one of its leaves. We
proved that every tree T has the leaf property when $r\leq 4$, and in 2017 Borg
and other authors gave examples of families of trees not having the leaf
property when $r\geq 5$. A split vertex in a tree is a vertex of degree at
least 3. A spider is a tree with exactly one split vertex. Here we prove that
all spiders have the leaf property for all $r\leq \alpha(G)$, where $\alpha(G)$
is the independence number of $G$, and we characterize which of its leaves are
maximum star centers. A pendant tree is one for which each of its split
vertices is adjacent to a leaf. Here we show that all pendant trees have the
leaf property for all $r\leq \alpha(G)$. We also consider pendant trees with
exactly two split vertices and provide partial results on the locations of
their maximum star centers.
