  For certain random variables that arise as limits of functionals of random
finite trees, we obtain precise asymptotics for the logarithm of the right-hand
tail. Our results are based on the facts (i) that the random variables we study
can be represented as functionals of a Brownian excursion and (ii) that a large
deviation principle with good rate function is known explicitly for Brownian
excursion. Examples include limit distributions of the total path length and of
the Wiener index in conditioned Galton-Watson trees (also known as simply
generated trees). In the case of Wiener index (where we recover results proved
by Svante Janson and Philippe Chassaing by a different method) and for some
other examples, a key constant is expressed as the solution to a certain
optimization problem, but the constant's precise value remains unknown.
