The Continuum Random Tree III

Let (R(k), k ≥ 1) be random trees with k leaves, satisfying a consistency condition: Removing a random leaf from R(k) gives R(k - 1). Then under an extra condition, this family determines a random continuum tree L, which it is convenient to represent as a random subset of l₁. This leads to an abstract notion of convergence in distribution, as n → ∞, of (rescaled) random trees Jₙ on n vertices to a limit continuum random tree L. The notion is based upon the assumption that, for fixed k, the subtrees of Jₙ determined by k randomly chosen vertices converge to R(k). As our main example, under mild conditions on the offspring distribution, the family tree of a Galton-Watson branching process, conditioned on total population size equal to n, can be rescaled to converge to a limit continuum random tree which can be constructed from Brownian excursion.