  In this article, we will prove that if $G$ is a connected claw-free graph and
either $\sigma_6(G)\geq |G|-5$ or $\sigma_7(G)\geq |G|-2$, here $\sigma_k(G)$
is the minimmum degree sum of $k$ independent vertices in $G$, then $G$ has a
spanning tree with at most two branch vertices.
