## The continuum random tree. III.(English)Zbl 0791.60009

Summary: [For parts I and II see author, ibid. 19, No. 1, 1-28 (1991; Zbl 0722.60013) and the paper reviewed above.]
Let $$({\mathcal R} (k)$$, $$k \geq 1)$$ be random trees with $$k$$ leaves, satisfying a consistency condition: Removing a random leaf from $${\mathcal R} (k)$$ gives $${\mathcal R} (k-1)$$. Then under an extra condition, this family determines a random continuum tree $${\mathcal S}$$, which it is convenient to represent as a random subset of $$l_ 1$$. This leads to an abstract notion of convergence in distribution, as $$n \to \infty$$, of (rescaled) random trees $${\mathcal T}_ n$$ on $$n$$ vertices to a limit continuum random tree $${\mathcal S}$$. The notion is based upon the assumption that, for fixed $$k$$, the subtrees of $${\mathcal T}_ n$$ determined by $$k$$ randomly chosen vertices converge to $${\mathcal R} (k)$$. 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.

### MSC:

 60C05 Combinatorial probability 60B10 Convergence of probability measures 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 05C05 Trees 05C80 Random graphs (graph-theoretic aspects)

### Citations:

Zbl 0791.60008; Zbl 0722.60013
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