  We determine, to within O(1), the expected minimal position at level n in
certain branching random walks. The walks under consideration have displacement
vector (v_1,v_2,...), where each v_j is the sum of j independent Exponential(1)
random variables and the different v_i need not be independent. In particular,
our analysis applies to the Poisson-Dirichlet branching random walk and to the
Poisson-weighted infinite tree. As a corollary, we also determine the expected
height of a random recursive tree to within O(1).
