Brownian excursions and random trees

Abstract: Consider a function f∈ C_c(ℝ→ℝ₊)​​ which is not identically 0. We generalize the definition of a tree in such a way that we can both topologize the space of all trees and that supp f/∼_f​​ becomes a compact tree. If instead of a deterministic f​​ we consider a Brownian excursion, would there be a way to determine how the random tree supp B^ex/∼_B^ex​​ ``looks" like? Large random trees give us a way to answer this question. This talk will formalize some of these vague concepts and outline some of the proofs in this area.