For certain random variables that arise as limits of functionals of randomfinite trees, we obtain precise asymptotics for the logarithm of the right-handtail. Our results are based on the facts (i) that the random variables we studycan be represented as functionals of a Brownian excursion and (ii) that a largedeviation principle with good rate function is known explicitly for Brownianexcursion. Examples include limit distributions of the total path length and ofthe Wiener index in conditioned Galton-Watson trees (also known as simplygenerated trees). In the case of Wiener index (where we recover results provedby Svante Janson and Philippe Chassaing by a different method) and for someother examples, a key constant is expressed as the solution to a certainoptimization problem, but the constant's precise value remains unknown.
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