  We establish lower tail bounds for the height, and upper tail bounds for the
width, of critical size-conditioned Bienaym\'e trees. Our bounds are optimal at
this level of generality. We also obtain precise asymptotics for offspring
distributions within the domain of attraction of a Cauchy distribution, under a
local regularity assumption. Finally, we pose some questions on the possible
asymptotic behaviours of the height and width of critical size-conditioned
Bienaym\'e trees.
