  We study the structure of extreme level sets of a standard one dimensional
branching Brownian motion, namely the sets of particles whose height is within
a fixed distance from the order of the global maximum. It is well known that
such particles congregate at large times in clusters of order-one genealogical
diameter around local maxima which form a Cox process in the limit. We add to
these results by finding the asymptotic size of extreme level sets and the
typical height and shape of those clusters which carry such level sets. We also
find the right tail decay of the distribution of the distance between the two
highest particles. These results confirm two conjectures of Brunet and Derrida,
2011. The proofs rely on studying the cluster distribution and should carry
over to the branching random walk and the two-dimensional discrete Gaussian
free field with no conceptual difficulty.
