  We study the limiting extremal and cluster point processes of branching
Brownian motion. The former records the heights of all extreme values of the
process, while the latter records the relative heights of extreme values in a
genealogical neighborhood of order unity around a local maximum thereof. For
the extremal point process, we show that the mass of upper level sets $[-v,
\infty)$ grows as $C_\star Z v e^{\sqrt{2} v}(1+o(1))$ as $v \to \infty$,
almost surely, where $Z$ is the limit of the associated derivative martingale
and $C_\star \in (0, \infty)$ is a universal constant. For the cluster point
process, we show that the logarithm of the mass of $[-v, \infty)$ grow as
$\sqrt{2}v$ minus random fluctuations of order $v^{2/3}$, which are governed by
an explicit law in the limit. The first result improves upon the works of
Cortines et al. (arXiv:1703.06529) and Mytnik et al. (arXiv:2009.02042) in
which asymptotics are shown in probability, while the second makes rigorous the
derivation in the physics literature by Mueller et al. (arXiv:1910.06382) and
Le et al. (arXiv:2207.07672) and resolves a conjecture thereof.
