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The continuum random tree. II: An overview. (English) Zbl 0791.60008
Stochastic analysis, Proc. Symp., Durham/UK 1990, Lond. Math. Soc. Lect. Note Ser. 167, 23-70 (1991).
[For the entire collection see Zbl 0733.00017.]
[For part I see author, Ann. Probab. 19, No. 1, 1-28 (1991; Zbl 0722.60013).]
This paper is concerned with models of random trees on $$n$$ vertices in which the average distance between pairs of vertices grows as order $$n^{1/2}$$. The author presents an extensive review on old and recent results about random trees equivalent to Galton-Watson branching processes conditioned on the total population size equal to $$n$$. Such models of trees, after appropriate rescaling the edges, converge in distribution as $$n \to \infty$$ to the so-called compact continuum random tree which is closely connected with the Brownian excursion and the 3- dimensional Bessel process. Limit theorems of such type are discussed in detail. Finally, the author proposes models where the limit trees obtained are considered as variants of a superprocess. [For part III see below].

##### MSC:
 60C05 Combinatorial probability 05C05 Trees 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 05C80 Random graphs (graph-theoretic aspects)