Partition-valued and measure-valued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes x and y runs the risk of a binary collision to form a single mass of magnitude x+y at rate (x; y), for some non-negative, symmetric collision rate kernel (x; y). Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable choice of state space, and under appropriate restrictions on and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Feller-like processes. A number of further results are obtained for the additive coalescent with collision kernel (x; y) = x + y. This process, which arises fro...

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...

Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...

Extensions of binomial and multinomial formulae due to Abel, Cayley and Hurwitz are related to the probability distributions of various random subsets, trees, forests, and mappings. For instance, an extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is defined by terms of a multinomial expansion over rooted labeled forests which generalizes Cayley's expansion over unrooted labeled trees. Contents 1 Introduction 2 Research supported in part by N.S.F. Grant DMS97-03961 2 Probabilistic Interpretations 5 3 Cayley's multinomial expansion 11 4 Random Mappings 14 4.1 Mappings from S to S : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 4.2 The random set of cyclic points : : : : : : : : : : : : : : : : : : : : : : : 18 5 Random Forests 19 5.1 Distribution of the roots of a p-forest : : : : : : : : : : : : : : : : : : : : 19 5.2 Conditioning on the set...

Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions due to Abel, Cayley and Hurwitz. Relations between these combinatorial objects, such as Joyal's bijection between mappings and marked rooted trees, have interesting probabilistic interpretations, and applications to the asymptotic structure of large random trees and mappings. An extension of Hurwitz's binomial formula is associated with the probability distribution of the random set of vertices of a fringe subtree in a random forest whose distribution is defined by terms of a multinomial expansion over rooted labeled forests. Research supported in part by N.S.F. Grants DMS 97-03961 and DMS-0071448 1 Contents 1

We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such asdegree sequence, shape, and total length are derived. An interpretation is given in terms of sampling from the inhomogeneous continuum random tree of Aldous and Pitman (1998). Key words and phrases. Continuum tree, enumeration, random tree, spanning tree, weighted tree, Cayley's multinomial expansion.

For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a p-forest if given F has m edges the vector of out-degrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 S), and given also these out-degrees the distribution of F is uniform on all forests with the given out-degrees. The family of distributions of p-forests is studied, and shown to be closed under various operations involving deletion of edges. Some related enumerations of rooted labeled forests are obtained as corollaries. 1 Introduction Let F(S) denote the set of all forests of rooted trees labeled by a finite set S of size jSj. Each f 2 F(S) is a directed graph labeled by S, that is a subset of S \Theta S, such that each Research supported in part by N.S.F. Grant DMS97-03961 connected component of the graph is a tree with edges directed away from some root vertex. The notation v f ...

Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of non-root vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution is described in a limiting regime where the distribution of the delabeled fringe subtree approaches that of a Galton-Watson tree with a mixed Poisson offspring distribution. 1 Introduction and statement of results Hurwitz [10] discovered the following identity of polynomials in n + 2 variables x; y and z s ; s 2 [n] := f1; : : : ; ng, which reduces to the binomial expansion of (x + y) n when Research supported in part by N.S.F. Grant DMS97-03961 z s j 0: X A`[n] x(x + z A ) jAj\Gamma1 (y + z ¯ A ) j ¯ Aj = (x + y + z [n] ) n (1) where the sum is over all 2 n subsets A of [n], with the notations z A := P s2A z s , and jAj for the number of elements of A, and ¯ A := [n] \...

This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ` S the polynomial enumerating out-degrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is ( P r2R x r )( P s2S x s ) jS j\GammajRj\Gamma1

Dedicated to T. GALLAI. at the occasion of his 50th birthday