Random recursive trees exhibit polynomial upper-tail and stretched-exponential lower-tail large deviation probabilities for height, with an atypical prefactor in the lower tail that grows slower than any n-fold iterated logarithm.
Correction terms for the height of weighted recursive trees
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Atypical Decay Rates for Atypical Heights in Random Recursive Trees
Random recursive trees exhibit polynomial upper-tail and stretched-exponential lower-tail large deviation probabilities for height, with an atypical prefactor in the lower tail that grows slower than any n-fold iterated logarithm.