On trees invariant under edge contraction

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.