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In this paper, we prove that every $n$-vertex connected $K_{1,5}$-free graph$G$ with $\sigma_4(G)\geq n-1$ contains a spanning tree with at most $5$ leavesand branch vertices in total. Moreover, the degree sum condition"$\sigma_4(G)\geq n-1$" is best possible.

Spanning trees with at most $5$ leaves and branch vertices in total of $K_{1,5}$-free graphs