ON NODES OF GIVEN DEGREE IN RANDOM TREES
Michael Drmota · 1997
. Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. We discuss the process L (d) n (t) which is then number of nodes of degree d in layer t. It is shown that the process n 1=2 L (d) n (n 1=2 t) converges weakly to Brownian excursion local time. This is done via characteristic functions which are obtained by means of generating functions arising from the combinatorial setup and complex contour integration. 1. Introduction Consider a class A of plane rooted trees. Dene for each T 2 A the size jT j by the number of nodes T consists of and a weight !(T ) = Y k0 ' nk (T ) k ; (1.1) where (' k ; k 0) are non-negative numbers and n k (T ) is the number of nodes v 2 T with out-degree k. Furthermore set an = X T :jT j=n !(T ): Then the corresponding generating function (GF) a(z) = P n0 an z n satises the functional equation a(z) = z'(a(z)); (1.2) where '(t) = P k0 '...