Fringe subtrees of random trees
Consider a rooted tree. For each node in the tree, consider the subtree rooted at that node, i.e., the node and its descendents. These subtrees are called fringe subtrees (Aldous, 1991). We are interested in the collection of all fringe subtrees of a random tree, for example a Galton-Watson tree, and in particular the number of fringe subtrees equal (i.e. isomorphic) to a given tree. We prove asymptotic normality in some situations, but other cases are still open.