  In this paper, we study uniform rooted plane trees with given degree
sequence. We show, under some natural hypotheses on the degree sequence, that
these trees converge toward the so-called Inhomogeneous Continuum Random Tree
after renormalisation. Our proof relies on the convergence of a modification of
the well-known Lukasiewicz path. We also give a unified treatment of the limit,
as the number of vertices tends to infinity, of the fragmentation process
derived by cutting-down the edges of a tree with a given degree sequence,
including its geometric representation by a lamination-valued process. The
latter is a collection of nested laminations that are compact subsets of the
unit disk made of non-crossing chords. In particular, we prove an equivalence
between Gromov-weak convergence of discrete trees and the convergence of their
associated lamination-valued processes.
