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In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$with $\sigma_4(G)\geq n-1$ contains a spanning tree with at most $3$ leaves. Inthis paper, we prove an analogue of Kyaw's result for connected $K_{1,5}$-freegraphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with$\sigma_5(G)\geq n-1$ contains a spanning tree with at most $4$ leaves.Moreover, the degree sum condition `$\sigma_5(G)\geq n-1$' is best possible.

Spanning trees with at most 4 leaves in $K_{1,5}-$free graphs