I will speak about results from an ongoing project with Louigi Addario-Berry. I will present non-asymptotic, universal height bounds for random combinatorial trees. We use these results to obtain new height bounds on conditioned Bienaymé-Galton-Watson trees and simply generated trees. Moreover, I will introduce a stochastic domination result for combinatorial trees that implies that binary trees are stochastically the tallest. These results are based on a new bijection between trees and sequences that was introduced in a joint work with Louigi Addario-Berry, Mickaël Maazoun and James Martin.