## Enumerations Of Trees And Forests Related To Branching Processes And Random Walks (1997)

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Venue: | Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci |

Citations: | 38 - 15 self |

### BibTeX

@INPROCEEDINGS{Pitman97enumerationsof,

author = {Jim Pitman},

title = {Enumerations Of Trees And Forests Related To Branching Processes And Random Walks},

booktitle = {Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci},

year = {1997},

pages = {163--180}

}

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### Abstract

In a Galton-Watson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined 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 n\Gammak\Gamma1 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 formula, which is then obtained as a byproduct. This approach unifies an...