  A $k$-universal permutation, or $k$-superpermutation, is a permutation that
contains all permutations of length $k$ as patterns. The problem of finding the
minimum length of a $k$-superpermutation has recently received significant
attention in the field of permutation patterns. One can ask analogous questions
for other classes of objects. In this paper, we study $k$-supertrees. For each
$d\geq 2$, we focus on two types of rooted plane trees called $d$-ary plane
trees and $[d]$-trees. Motivated by recent developments in the literature, we
consider "contiguous" and "noncontiguous" notions of pattern containment for
each type of tree. We obtain both upper and lower bounds on the minimum
possible size of a $k$-supertree in three cases; in the fourth, we determine
the minimum size exactly. One of our lower bounds makes use of a recent result
of Albert, Engen, Pantone, and Vatter on $k$-universal layered permutations.
