Bounded branching process and and/or tree evaluation

We study the tail distribution of supercritical branching processes for which the number of offspring of an element is bounded. Given a supercritical branching process { Z n }   0 ∞with a bounded offspring distribution, we derive a tight bound, decaying super‐exponentially fast as c increases, on the probability Pr[ Z n > cE ( Z n )], and a similar bound on the probability Pr[ Z n ≤ E(Z n )/c ] under the assumption that each element generates at least two offspring. As an application, we observe that the execution of a canonical algorithm for evaluating uniform AND/OR trees in certain probabilistic models can be viewed as a two‐type supercritical branching process with bounded offspring, and show that the execution time of this algorithm is likely to concentrate around its expectation, with a standard deviation of the same order as the expectation.