A tree with atmost m leaves is called an m-ended tree. Kyaw proved that every connected K
1,4-free graph with σ
4(G) ⩽ n−1 contains a spanning 3-ended tree. In this paper we obtain a result for k-connected K
1,4-free graphs with k ⩽ 2. Let G be a k-connected K
1,4-free graph of order n with k ⩽ 2. If σ
k+3(G) ⩽ n+2k −2, then G contains a spanning 3-ended tree.
