  We consider a branching random walk for which the maximum position of a
particle in the n'th generation, M_n, has zero speed on the linear scale: M_n/n
--> 0 as n --> infinity. We further remove ("kill") any particle whose
displacement is negative, together with its entire descendence. The size $Z$ of
the set of un-killed particles is almost surely finite. In this paper, we
confirm a conjecture of Aldous that Exp[Z] < infinity while Exp[Z log
Z]=infinity. The proofs rely on precise large deviations estimates and ballot
theorem-style results for the sample paths of random walks.
