Moments of Moments and Branching Random Walks

We calculate, for a branching random walk [Formula: see text] to a leaf l at depth n on a binary tree, the positive integer moments of the random variable [Formula: see text] , for [Formula: see text] . We obtain explicit formulae for the first few moments for finite n. In the limit [Formula: see text] , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.