  A fringe subtree of a rooted tree is a subtree induced by one of the vertices
and all its descendants. We consider the problem of estimating the number of
distinct fringe subtrees in two types of random trees: simply generated trees
and families of increasing trees (recursive trees, $d$-ary increasing trees and
generalized plane-oriented recursive trees). We prove that the order of
magnitude of the number of distinct fringe subtrees (under rather mild
assumptions on what `distinct' means) in random trees with $n$ vertices is
$n/\sqrt{\log n}$ for simply generated trees and $n/\log n$ for increasing
trees.
