Enumerations of trees and forests related to branching processes and random walks
نویسنده
چکیده
In a Galton-Watson branching process with o spring distribution (p0; p1; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the rst passage time to k for a random walk started at 0 which takes steps of size j with probability pj+1 for j 1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coe cients in the power series expansion of f(z) in terms of those of g(z) for f(z) de ned implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn k 1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange
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