## Probability Surveys

### Fringe trees, Crump–Mode–Jagers branching processes and $m$-ary search trees

#### Abstract

This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump–Mode–Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) $m$-ary search trees, as well as some other classes of random trees.

We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of $m$-ary search trees in detail; this seems to be new.

Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for $m$-ary search trees, and give for example new results on protected nodes in $m$-ary search trees.

A separate section surveys results on the height of the random trees due to Devroye (1986), Biggins (1995, 1997) and others.

This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees.

#### Article information

Source
Probab. Surveys, Volume 14 (2017), 53-154.

Dates
First available in Project Euclid: 22 March 2017

https://projecteuclid.org/euclid.ps/1490169611

Digital Object Identifier
doi:10.1214/16-PS272

Mathematical Reviews number (MathSciNet)
MR3626585

Zentralblatt MATH identifier
06702348

#### Citation

Holmgren, Cecilia; Janson, Svante. Fringe trees, Crump–Mode–Jagers branching processes and $m$-ary search trees. Probab. Surveys 14 (2017), 53--154. doi:10.1214/16-PS272. https://projecteuclid.org/euclid.ps/1490169611

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