  If trees are constructed from a pure birth process and one defines the depth
of a leaf to be the number of edges to its root, it is known that the variance
in the depth of a randomly selected leaf of a randomly selected tree grows
linearly in time. In this letter, we instead consider the variance of the
average depth of leaves within each individual tree, establishing that, in
contrast, it converges to a constant, $7$. This result indicates that while the
variance in leaf depths amongst the ensemble of pure birth processes undergoes
large fluctuations, the average depth across individual trees is much more
consistent.
