  We study the conditions for families of subtrees to exist with high
probability (whp) in a Galton-Walton tree of size $n$. We first give a Poisson
approximation of fringe subtree counts, which yields the height of the maximal
complete $r$-ary fringe subtree. Then we determine the maximal $K_n$ such that
every tree of size at most $K_n$ appears as fringe subtree whp. Finally, we
study non-fringe subtree counts and determine the height of the maximal
complete $r$-ary non-fringe subtree.
