Dependence between path-length and size in random digital trees

Michael Fuchs*, Hsien Kuei Hwang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

Original languageEnglish
Pages (from-to)1125-1143
Number of pages19
JournalJournal of Applied Probability
Issue number4
StatePublished - 1 Dec 2017


  • asymptotic normality
  • contraction method
  • covariance
  • de-Poissonization
  • integral transform
  • Pearson's correlation coefficient
  • Poissonization
  • Random tries
  • total path length

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