Bimodality and Phase Transitions in the Profile Variance of Random Binary Search Trees

We show that the variances of the profile (number of nodes at each level) of random binary search trees undergoes asymptotically four phase transitions and exhibits a bimodal or "two-humped" behavior, in contrast to the unimodality of the expected value of the profiles. Precise asymptotic approximations are derived. The same types of phenomena also hold for the profile of random recursive trees.

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