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

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 height, saturation level, typical depth and total path length, 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.