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This survey studies asymptotics of random fringe trees and extended fringetrees in random trees that can be constructed as family trees of aCrump-Mode-Jagers branching process, stopped at a suitable time. This includesrandom recursive trees, preferential attachment trees, fragmentation trees,binary search trees and (more generally) $m$-ary search trees, as well as someother classes of random trees.  We begin with general results, mainly due to Aldous (1991) and Jagers andNerman (1984). The general results are applied to fringe trees and extendedfringe trees for several particular types of random trees, where the theory isdeveloped in detail. In particular, we consider fringe trees of $m$-ary searchtrees in detail; this seems to be new.  Various applications are given, including degree distribution, protectednodes and maximal clades for various types of random trees. Again, we emphasiseresults for $m$-ary search trees, and give for example new results on protectednodes in $m$-ary search trees.  A separate section surveys results on height, saturation level, typical depthand total path length, due to Devroye (1986), Biggins (1995, 1997) and others.  This survey contains well-known basic results together with some additionalgeneral results as well as many new examples and applications for variousclasses of random trees.

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