Total Path Length For Random Recursive Trees

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 (1991) 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 W d = U(1 + W ) + (1− U)W ∗ − E(U), where E(x) := −x lnx − (1 − 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. Simulations exhibit a high degree of accuracy in the approximation. Research for the first author supported by NSF grant DMS-9626597. Research for the second author supported by NSF grant DMS-9626756. AMS 1991 subject classifications. Primary 05C05, 60C05; secondary 60F25.