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Using the CLEO II detector operating at the CESR e+e- collider, we have measured the structure functions in the decay tau+/- --> pi+/- pi0 pi0 nu, based on a sample corresponding to 4*10E6 produced tau-pair events. We determine the integrated structure functions, which depend only on the three pion invariant mass, as well as the structure functions differential in the Dalitz plot. We extract model independent limits on non-axial-vector contributions from the measured structure functions as less than 16.6% of the total branching fraction, at the 95% confidence level. Separating the non-axial-vector contributions into scalar and vector contributions, we measure that scalars (vectors) contribute with less than 9.4% (7.3%) to the total branching ratio, at the 95% confidence level.