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Total Path Length for Random Recursive Trees

Published online by Cambridge University Press:  01 July 1999

Division of Mathematics and Computer Science, Truman State University, Kirksville, MO 63501-4221, USA (e-mail:
Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218-2682, USA (e-mail:


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 [10] showed that Wn := (TnE[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

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.

Research Article
1999 Cambridge University Press

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