Tight universal bounds on the height times the width of random trees

We obtain assumption-free, non-asymptotic, uniform bounds on the product of the height and the width of uniformly random trees with a given degree sequence, conditioned Bienaymé trees and simply generated trees. We show that for a tree of size $n$, this product is $O(n\log n)$ in probability, answering a question by Addario-Berry (2019). The order of this bound is tight in this generality.