Profile of Random Exponential Binary Trees

  • Yarong Feng
  • Hosam Mahmoud


We introduce the random exponential binary tree (EBT) and study its profile. As customary, the tree is extended by padding each leaf node (considered internal), with the appropriate number of external nodes, so that the outdegree of every internal node is made equal to 2. In a random EBT, at every step, each external node is promoted to an internal node with probability p, stays unchanged with probability 1 - p, and the resulting tree is extended. We study the internal and external profiles of a random EBT and get exact expectations for the numbers of internal and external nodes at each level. Asymptotic analysis shows that the average external profile is richest at level \(\frac {2p}{p+1}n\), and it experiences phase transitions at levels a n, where the a’s are the solutions to an algebraic equation. The rates of convergence themselves go through an infinite number of phase changes in the sublinear range, and then again at the nearly linear levels.


Binary tree Random graph Internal (external) profile Asymptotic analysis Phase transition Two-dimensional recurrence Generating function 

Mathematics Subject Classification (2010)

Primary: 90B15; Secondary: 60C05 05C80 65Q30 


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of StatisticsThe George Washington UniversityWashingtonUSA

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