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 ln x - (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 ...

• Type: Journal
• Year: 1999
• Where: CPC
• Authors: Robert P. Dobrow, James Allen Fill
• Keywords: CPC 1999, Recursive Trees, Search Cost, Total Path Length