  Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of
degree at least three is a branch vertex of $T$. For two distinct vertices
$u,v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and
$v.$ For a leaf $x$ of $T$, let $y_x$ denote the nearest branch vertex to $x$.
For every leaf $x$ of $T$, we remove the path $P_T [x, y_x)$ from $T$, where
$P_T [x, y_x)$ denotes the path connecting $x$ to $y_x$ in $T$ but not
containing $y_x$. The resulting subtree of $T$ is called the {\it reducible
stem } of $T$. In this paper, we first use a new technique of Gould and Shull
to state a new short proof for a result of Kano et al. on the spanning tree
with a bounded number of leaves in a claw-free graph. After that, we use that
proof to give a sharp sufficient condition for a claw-free graph having a
spanning tree whose reducible stem has few leaves.
