  We use a new stick-breaking construction, also introduced independently by
Addario-Berry, Donderwinkel, Maazoun, and Martin in arXiv:2107.09726, to study
uniform rooted trees with fixed degree sequence D (D-trees). This construction
allows us to prove, under natural conditions of convergence of the degree
sequence, that D-trees converge toward either P-trees or ICRT. Alongside we
provide an upper-bound for the height of D-trees. We deduce similar results
concerning P-trees and ICRT. In passing we confirm a conjecture of Aldous,
Miermont, and Pitman arXiv:math/0401115 stating that L\'evy trees are ICRT with
random parameters.
