Limit of trees with fixed degree sequence

Abstract

We show, under natural conditions, that uniform rooted trees with fixed degree sequence converge after renormalization toward inhomogeneous continuum random trees (ICRT). We also provide a sharp upper-bound for the tail of their heights. We also extend our results to P-trees, ICRT, and trees with random degree sequence. In passing we confirm a conjecture of Aldous, Miermont, and Pitman stating that L\'evy trees are ICRT with random parameters.