We examine relations for hydraulic geometry of alluvial, single-thread gravel bed rivers with definable bankfull geometries. Four baseline data sets determine relations for bankfull geometry, i.e., bankfull depth, bankfull width, and down-channel slope as functions of bankfull discharge and bed surface median sediment size. These relations show a considerable degree of universality. This universality applies not only within the four sets used to determine the forms but also to three independent data sets as well. We study the physical basis for this universality in terms of four relations, the coefficients and exponents of which can be back calculated from the data: (1) a Manning-Strickler-type relation for channel, resistance, (2) a channel-forming relation expressed in terms of the ratio of bankfull Shields number to critical Shields number, (3) a relation for critical Shields number as a function of dimensionless discharge, and (4) a "gravel yield" relation specifying the (estimated) gravel transport rate at bankfull flow as a function of bankfull discharge and gravel size. We use these underlying relations to explore why the dimensionless bankfull relations are only quasi-universal and to quantify the degree to which deviation from universality can be expected. The analysis presented here represents an alternative to extremal formulations to predict hydraulic geometry.