The Continuum Random Tree III

Abstract

Let $(\mathscr{R}(k), k \geq 1)$ be random trees with $k$ leaves, satisfying a consistency condition: Removing a random leaf from $\mathscr{R}(k)$ gives $\mathscr{R}(k - 1)$. Then under an extra condition, this family determines a random continuum tree $\mathscr{L}$, which it is convenient to represent as a random subset of $l_1$. This leads to an abstract notion of convergence in distribution, as $n \rightarrow \infty$, of (rescaled) random trees $\mathscr{J}_n$ on $n$ vertices to a limit continuum random tree $\mathscr{L}$. The notion is based upon the assumption that, for fixed $k$, the subtrees of $\mathscr{J}_n$ determined by $k$ randomly chosen vertices converge to $\mathscr{R}(k)$. 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.