  For an integer $k\geq 2$, a spanning tree of a graph without no vertices of
degree from $2$ to $k$ is call a {\it $[2,k]$-ST} of the graph. The concept of
$[2,k]$-STs is a natural extension of a homeomorphically irreducible spanning
tree (or HIST), which has been a well-studied graph-structure. In this paper,
we give a new strategy for finding $[2,k]$-STs. By using the strategy, we
refine or extend a known degree-sum condition for the existence of a HIST.
Furthermore, we also investigate on a degree-product condition for the
existence of a $[2,k]$-ST.
