In this article, we will prove that if $G$ is a connected claw-free graph and either $σ_6(G)\geq |G|-5$ or $σ_7(G)\geq |G|-2$, here $σ_k(G)$ is the minimmum degree sum of $k$ independent vertices in $G$, then $G$ has a spanning tree with at most two branch vertices.
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