  We give new results on the growth of the number of particles in a dyadic
branching Brownian motion which follow within a fixed distance of a path
$f:[0,\infty)\to \mathbb{R}$. We show that it is possible to count the number
of particles without rescaling the paths. Our results reveal that the number of
particles along certain paths can oscillate dramatically. The methods used are
entirely probabilistic, taking advantage of the spine technique developed by,
amongst others, Lyons et al, Kyprianou, and Hardy & Harris.
