We study the diameter of L{\'e}vy trees that are random compact metric spaces
obtained as the scaling limits of Galton-Watson trees. L{\'e}vy trees have been
introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum
Random Tree (1991) that corresponds to the Brownian case. We first characterize
the law of the diameter of L{\'e}vy trees and we prove that it is realized by a
unique pair of points. We prove that the law of L{\'e}vy trees conditioned to
have a fixed diameter r ∈ (0, ∞) is obtained by glueing at their
respective roots two independent size-biased L{\'e}vy trees conditioned to have
height r/2 and then by uniformly re-rooting the resulting tree; we also
describe by a Poisson point measure the law of the subtrees that are grafted on
the diameter. As an application of this decomposition of L{\'e}vy trees
according to their diameter, we characterize the joint law of the height and
the diameter of stable L{\'e}vy trees conditioned by their total mass; we also
provide asymptotic expansions of the law of the height and of the diameter of
such normalised stable trees, which generalises the identity due to Szekeres
(1983) in the Brownian case
