A line-breaking construction of the stable trees

Abstract

We give a new, simple construction of the a α stable tree for α ∈ (1, 2] . We obtain it as the closure of an increasing sequence of R-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing ℝ+-valued Markov chain. For ∈ = 2, we recover Aldous’ line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.