  In this article, we prove global and local (point-wise) volume and heat
kernel bounds for the continuum random tree. We demonstrate that there are
almost-surely logarithmic global fluctuations and log-logarithmic local
fluctuations in the volume of balls of radius $r$ about the leading order
polynomial term as $r\to0$. We also show that the on-diagonal part of the heat
kernel exhibits corresponding global and local fluctuations as $t\to0$
almost-surely. Finally, we prove that this quenched (almost-sure) behaviour
contrasts with the local annealed (averaged over all realisations of the tree)
volume and heat kernel behaviour, which is smooth.
