  In a recent pair of papers Gorin and Shkolnikov (2018) and Hariya (2016) have
shown that the area under normalized Brownian excursion minus one half the
integral of the square of its total local time is a centered normal random
variable with variance $\frac{1}{12}$. Gaudreau Lamarre and Shkolnikov (2019)
generalized this to Brownian bridges, and ask for a combinatorial
interpretation. We provide a combinatorial interpretation using random forests
on $n$ vertices. In particular, we show that there is a process level
generalization for a certain infinite forest model. We also show analogous
results for a variety of other related models using stochastic calculus.
