  In this paper, we study averaging operators from an algebraic and
combinatorial point of view. We first construct free averaging algebras in
terms of a class of bracketed words called averaging words. We next apply this
construction to obtain one and two variable generating functions for subsets of
averaging words when the averaging operator is taken to be idempotent. When the
averaging algebra has an idempotent generator, the generating function in one
variable is twice the generating function for large Schr\"oder numbers, leading
us to give interpretations of large Schr\"oder numbers in terms of bracketed
words and rooted trees, as well as a recursive formula for these numbers. We
also give a representation of free averaging algebras by unreduced trees and
apply it to give a combinatorial description of the operad of averaging
algebras.
