  In 1998, Broersma and Tuinstra [J. Graph Theory \textbf{29} (1998), 227-237]
proved that if $G$ is a connected graph satisfying $\sigma_2(G) \geq |G|-k+1$
then $G$ has a spanning $k-$ended tree. They also gave an example to show that
the condition "$\sigma_2(G) \geq |G|-k+1$" is sharp. In this paper, we
introduce a new progress for this result. Let $K_{m,m+k}$ be a complete
bipartite graph with bipartition $V(K_{m,m+k})=A\cup B, |A|=m, |B|=m+k.$ Denote
by $H$ to be the graph obtained from $K_{m,m+k}$ by adding (or no adding) some
edges with two end vertices in $A.$ We prove that if $G$ is a connected graph
satisfying $\sigma_2(G) \geq |G|-k$ then $G$ has a spanning $k-$ended tree
except for the case $G$ is isomorphic to a graph $H.$ As a corollary of our
main result, a sufficient condition for a graph to have a few branch vertices
is given.
