  Over thirty years ago, Kalai proved a beautiful $d$-dimensional analog of
Cayley's formula for the number of $n$-vertex trees. He enumerated
$d$-dimensional hypertrees weighted by the squared size of their
$(d-1)$-dimensional homology group. This, however, does not answer the more
basic problem of unweighted enumeration of $d$-hypertrees, which is our concern
here. Our main result, Theorem 1.4, significantly improves the lower bound for
the number of $d$-hypertrees. In addition, we study a random $1$-out model of
$d$-complexes where every $(d-1)$-dimensional face selects a random $d$-face
containing it, and show it has a negligible $d$-dimensional homology.
