Detailed numerical calculations, which are supported experimentally, are used to investigate the sensitivity of flame structure to fuel premixedness and overall strain rate. The study covers a wide range of fuel premixedness-from phi = 1.3 to phi = 2.0, plus a pure diffusion flame-over a wide range of strain rates-from moderate to near extinction-in methane-air vs, air counter-flow flames. The flames are observed to change drastically in structure and character-from a single, merged flame in the vicinity of the stagnation plane to a double flame consisting of a premixed-type, fuel-side flame and a stagnation-region diffusion flame-as the fuel stream equivalence ratio is perturbed slightly below phi approximate to 1.5-1.4. Accordingly, the mode and amount of NOchi formation changes severely. This duality in flame structure is further discerned by monitoring the relative locations of CH and OH profiles, as these are indicators of specific dame chemistry. The exact value of the "changeover" equivalence ratio depends upon the flame's strain rate, and in fact, Barnes close enough to extinction remain as a merged flame structure even at the lowest equivalence ratio. The maximum fuel-side velocity gradient is shown to be an extremely sensitive and sharp indicator of flame character, being completely insensitivity to fuel-stream equivalence ratio above certain strain-dependent values, but varying sharply with equivalence ratio below these values. Other parameters, such as the width of the temperature or product species profiles, are shown to be indicators of flame structure, also, but are not nearly as sharply responsive as the fuel-side velocity gradient. These results could have important implications for design criteria for commercial burners, as well as for applications to the prediction of turbulent flame structure, including suppression and extinction.