  In a given graph, a HIST is a spanning tree without $2$-valent vertices.
Motivated by developing a better understanding of HIST-free graphs, i.e. graphs
containing no HIST, in this article's first part we study HIST-critical graphs,
i.e. HIST-free graphs in which every vertex-deleted subgraph does contain a
HIST (e.g. a triangle). We give an almost complete characterisation of the
orders for which these graphs exist and present an infinite family of planar
examples which are $3$-connected and in which nearly all vertices are
$4$-valent. This leads naturally to the second part in which we investigate
planar $4$-regular graphs with and without HISTs, motivated by a conjecture of
Malkevitch, which we computationally verify up to order $22$. First we
enumerate HISTs in antiprisms, whereafter we present planar $4$-regular graphs
with and without HISTs, obtained via line graphs.
