The Erdos-Sos conjecture for spiders of diameter 9

The Erdos-Sos conjecture says that any graph G on n vertices with e(G) > k-1/2n contains every tree of k edges. In J. Graph Theory 21 (1996), 229-234, Wozniak proved that the conjecture is true for spiders of diameter at most 4. In Discrete Math. (2007) (in press), Fan and Sun proved that the conjecture is true for spiders with no leg of length more than 4. In this paper, we prove that the conjecture is true for spiders of diameter at most 9, which strengthens both results mentioned above.