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This IMA Volume in Mathematics and its Applications CLASSICAL AND MODERN BRANCHING PROCESSES is based on the proceedings with the same title and was an integral part of the 1993-94 IMA program on "Emerging Applications of Probability." We would like to thank Krishna B. Athreya and Peter J agers for their hard work in organizing this meeting and in editing the proceedings. We also take this opportunity to thank the National Science Foundation, the Army Research Office, and the National Security Agency, whose financial support made this workshop possible. A vner Friedman Robert Gulliver v PREFACE The IMA workshop on Classical and Modern Branching Processes was held during June 13-171994 as part of the IMA year on Emerging Appli­ cations of Probability. The organizers of the year long program identified branching processes as one of the active areas in which a workshop should be held. Krish­ na B. Athreya and Peter Jagers were asked to organize this. The topics covered by the workshop could broadly be divided into the following areas: 1. Tree structures and branching processes; 2. Branching random walks; 3. Measure valued branching processes; 4. Branching with dependence; 5. Large deviations in branching processes; 6. Classical branching processes.

Large deviation rates for supercritical and critical branching processes.- How fast does a general branching random walk spread?.- Boltzmann-Gibbs weights in the branching random walk.- Stochastic monotonicity and branching processes.- Multilevel multitype branching models of an information system.- On the shape of the wavefront of branching random walk.- Limiting distributions in branching processes with two types of particles.- Depth―first search of random trees, and Poisson point processes.- Towards dependence in general branching processes.- A criterion of boundedness of discrete branching random walk.- Quasistationarity in a branching model of division-within-division.- Population and density dependent branching processes.- Directed polymers in random media and spin glass models on trees..- A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes.- On two measures defined on the boundary of a branching tree.- Which critically branching populations persist?.- A simple path to Biggins’ martingale convergence for branching random walk.- Unsolved problems concerning random walks on trees.- Branching processes with local dependencies.- Sharpness of second moment criteria for branching and tree-indexed processes.- On the recognition and structure of probability generating functions.- Record values of a family of branching processes.- Limit skeleton for critical crump-mode-Jagers branching processes.- Markov cascades.- Limit theorems for branching processes with random migration stopped at zero.