Total path length, or search cost, for a rooted tree is defined as the sum of all
root-to-node distances. Let Tn be the total path
length for a random recursive tree of order n. Mahmoud  showed that
Wn := (Tn −
E[Tn])/n converges almost surely
and in L2 to a nondegenerate limiting random variable W.
Here we give recurrence relations for the moments of Wn and of
W and show that Wn converges to W in
Lp for each 0 < p < ∞. We confirm the
conjecture that the distribution of W is not normal. We also show that the
distribution of W is characterized among all distributions having zero mean and finite
variance by the distributional identity
where [Escr ](x) := − x ln x −
(1 minus; x) ln(1 − x) is the binary entropy function, U
is a uniform (0, 1) random variable, W* and W have the same
distribution, and U, W and W* are mutually independent.
Finally, we derive an approximation for the distribution of W using a Pearson
curve density estimator.