The stabilization and extinction phenomena for a two-dimensional diffusion flame around the leading edge of an upright solid fuel under naturally convective environments are studied theoretically. The governing system consists of two-dimensional Navier-Stokes' momentum (including a body force term), energy, species, and continuity equations, together with a finite-rate chemical kinetics described by a second-order Arrhenius-type rate equation. The burning rate of fuel is expressed by an Arrhenius-type pyrolysis law. The nondimensional model is solved numerically. Two parametric Studies, based on changing gravity level and ambinet oxygen index, respectively, are investigated. Under the variation of gravity, both Damkohler number and preexponential factor for fuel pyrolysis are changed. Two quenching limits are identified separately at a very low and a high gravity level. The envelope flame exists when gravity is less than 1.95 g. As gravity is increased, there appears a transition to open-tip flame, stabilized on the side of fuel plate. A similar trend is found when ambient oxygen index is decreased from the higher value to the lower one but only with a quenching limit that is blowoff. The detailed flame structures for envelope and open-tip flames are illustrated graphically.