  We prove non-asymptotic stretched exponential tail bounds on the height of a
randomly sampled node in a random combinatorial tree, which we use to prove
bounds on the heights and widths of random trees from a variety of models. Our
results allow us to prove a conjecture and settle an open problem of Janson
(https://doi.org/10.1214/11-PS188), and nearly prove another conjecture and
settle another open problem from the same work (up to a polylogarithmic
factor).
  The key tool for our work is an equivalence in law between the degrees along
the path to a random node in a random tree with given degree statistics, and a
random truncation of a size-biased ordering of the degrees of such a tree. We
also exploit a Poissonization trick introduced by Camarri and Pitman
(https://doi.org/10.1214/EJP.v5-58) in the context of inhomogeneous continuum
random trees, which we adapt to the setting of random trees with fixed degrees.
  Finally, we propose and justify a change to the conventions of branching
process nomenclature: the name "Galton-Watson trees" should be permanently
retired by the community, and replaced with the name "Bienaym\'e trees".
