A spanning tree of a graph without no vertices of degree $2$ is called a {\it homeomorphically irreducible spanning tree} (or a {\it HIST}) of the graph. Albertson, Berman, Hutchinson and Thomassen~[J. Graph Theory {\bf 14} (1990), 247--258] gave a minimum degree condition for the existence of a HIST, and recently, Ito and Tsuchiya~[J. Graph Theory {\bf 99} (2022), 162--170] found a sharp degree-sum condition for the existence of a HIST. In this paper, we refine these results, and extend the first one to a spanning tree in which no vertex other than the endvertices has small degree.