Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T23:54:01.153Z Has data issue: false hasContentIssue false

Total Path Length for Random Recursive Trees

Published online by Cambridge University Press:  01 July 1999

ROBERT P. DOBROW
Affiliation:
Division of Mathematics and Computer Science, Truman State University, Kirksville, MO 63501-4221, USA (e-mail: [email protected])
JAMES ALLEN FILL
Affiliation:
Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218-2682, USA (e-mail: [email protected])

Abstract

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.

Type
Research Article
Copyright
1999 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)