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

Averaging algebras, Schr\"oder numbers, rooted trees and operads