  As a first step toward a characterization of the limiting extremal process of
branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math.
64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles
descend with overwhelming probability from ancestors having split either within
a distance of order 1 from time 0, or within a distance of order 1 from time
$t$. The result suggests that the extremal process of branching Brownian motion
is a randomly shifted cluster point process. Here we put part of this picture
on rigorous ground: we prove that the point process obtained by retaining only
those extremal particles which are also maximal inside the clusters converges
in the limit of large $t$ to a random shift of a Poisson point process with
exponential density. The last section discusses the Tidal Wave Conjecture by
Lalley and Sellke [Ann. Probab. 15 (1987) 1052-1061] on the full limiting
extremal process and its relation to the work of Chauvin and Rouault [Math.
Nachr. 149 (1990) 41-59] on branching Brownian motion with atypical
displacement.
