Spanning trees of a claw-free graph whose reducible stems have few leaves

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.