For a connected graph $G$, a spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST) if $T$ has no vertices of degree 2. Albertson {\em et al.} proved that it is $NP$-complete to decide whether a graph contains a HIST. In this paper, we provide some spectral conditions that guarantee the existence of a HIST in a connected graph. Furthermore, we also present some sufficient conditions in terms of the order of a graph $G$ to ensure the existence of a HIST in $G$.