  It is well-known that the height profile of a critical conditioned
Galton-Watson tree with finite offspring variance converges, after a suitable
normalization, to the local time of a standard Brownian excursion. In this
work, we study the distance profile, defined as the profile of all distances
between pairs of vertices. We show that after a proper rescaling the distance
profile converges to a continuous random function that can be described as the
density of distances between random points in the Brownian continuum random
tree.
  We show that this limiting function a.s. is H\"older continuous of any order
$\alpha<1$, and that it is a.e. differentiable. We note that it cannot be
differentiable at $0$, but leave as open questions whether it is Lipschitz, and
whether is continuously differentiable on the half-line $(0,\infty)$.
  The distance profile is naturally defined also for unrooted trees contrary to
the height profile that is designed for rooted trees. This is used in our
proof, and we prove the corresponding convergence result for the distance
profile of random unrooted simply generated trees. As a minor purpose of the
present work, we also formalize the notion of unrooted simply generated trees
and include some simple results relating them to rooted simply generated trees,
which might be of independent interest.
