Enumeration and randomized constructions of hypertrees

Over 30 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 that it has a negligible d ‐dimensional homology.