  The shrinking operation converts a hypergraph into a graph by choosing, from
each hyperedge, two endvertices of a corresponding graph edge. A hypertree is a
hypergraph which can be shrunk to a tree on the same vertex set.
Klimo\v{s}ov\'{a} and Thomass\'{e} [J. Combin. Theory Ser. B 156 (2022),
250--293] proved (as a tool to obtain their main result on edge-decompositions
of graphs into paths of equal length) that any rank $3$ hypertree $T$ can be
shrunk to a tree where the degree of each vertex is at least $1/100$ times its
degree in $T$. We prove a stronger and a more general bound, replacing the
constant $1/100$ with $1/2k$ when the rank is $k$. In place of entropy
compression (used by Klimo\v{s}ov\'{a} and Thomass\'{e}), we use a hypergraph
orientation lemma combined with a characterisation of edge-coloured graphs
admitting rainbow spanning trees.
