  We associate with a Bienayme-Galton-Watson branching process a family tree
rooted at the ancestor. For a positive integer N, define a complete N-ary tree
to be the family tree of a deterministic branching process with offspring
generating function s^N. We study the random variables V(N,n) and V(N) counting
the number of disjoint complete N-ary subtrees, rooted at the ancestor, and
having height n and infinity, respectively. Dekking (1991) and Pakes and
Dekking (1991) find recursive relations for Pr(V(N,n)>0) and Pr(V(N)>0)
involving the offspring probability generation function (pgf) and its
derivatives. We extend their results determining the probability distributions
of V(N,n) and V(N). It turns out that they can be expressed in terms of the
offspring pgf, its derivatives, and the above probabilities. We show how the
general results simplify in case of fractional linear, geometric, Poisson, and
one-or-many offspring laws.
