Large deviations for the leaves in some random trees
  Large deviation principles and related results are given for a class of Markov chains associated to the "leaves" in random recursive trees and preferential attachment random graphs, as well as the "cherries" in Yule trees.   In particular, the method of proof, combining analytic and Dupuis-Ellis type path arguments, allows for an explicit computation of the large deviation pressure. 
W. Bryc, D. Minda, S. Sethuraman