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CPC
1999

Total Path Length For Random Recursive Trees

11 years 7 months ago
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 ...
Robert P. Dobrow, James Allen Fill
Added 22 Dec 2010
Updated 22 Dec 2010
Type Journal
Year 1999
Where CPC
Authors Robert P. Dobrow, James Allen Fill
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