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We obtain new non-asymptotic tail bounds for the height of uniformly randomtrees with a given degree sequence, simply generated trees and conditionedBienaym\'e trees (the family trees of branching processes), in the processsettling two conjectures of Janson (2012) and answering several other questionsfrom the literature.  Moreover, we define a partial ordering on degree sequences and show that itinduces a stochastic ordering on the heights of uniformly random trees withgiven degree sequences. The latter result can also be used to show thatsub-binary random trees are stochastically the tallest trees with a givennumber of vertices and leaves (and thus that random binary trees are thestochastically tallest random homeomorphically irreducible trees with a givennumber of vertices).  Our proofs are based in part on a bijection introduced by Addario-Berry,Donderwinkel, Maazoun and Martin (2021), which can be recast to provide aline-breaking construction of random trees with given vertex degrees.

Random trees have height $O(\sqrt{n})$