  We study random trees which are invariant in law under the operation of
contracting each edge independently with probability $p\in(0,1)$. We show that
all such trees can be constructed through Poissonian sampling from a certain
class of random measured $\R$-trees satisfying a natural scale invariance
property. This has connections to exchangeable partially ordered sets,
real-valued self-similar increasing processes and quasi-stationary
distributions of Galton--Watson processes.
