Total Path Length for Random Recursive Trees

Abstract

Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let T_n be the total path length for a random recursive tree of order n. Mahmoud [10] showed that W_n := (T_n − E[T_n])/n converges almost surely and in L² to a nondegenerate limiting random variable W. Here we give recurrence relations for the moments of W_n and of W and show that W_n converges to W in L^p 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

formula here

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.