Numerical simulations of laminar two-dimensional triple flames are conducted to investigate the mechanisms of buoyancy-induced instabilities. These simulations are implemented for a selected range of gravity conditions and inlet scalar mixing widths for downward-propagating triple flames (propagating in the same direction as the gravity vector). Increases in the gravity force result in a transition from a stable to an unstable behavior, A linear inviscid stability analysis is performed to explore the mechanisms of instability and to estimate the most amplified frequencies. Unstable triple flame simulations provide detailed flow and scalar information for interrogating the mechanisms of buoyancy-induced instabilities in triple flames. These instabilities are accompanied by baroclinic generation of counter-vortices consistent with the Kelvin-Helmholtz instabilities. The computed onset of instabilities is accompanied by the advection of the triple flames downstream from their stabilization point. This advection plays a dominant role in the unstable behavior, further illustrating the hydrodynamic, buoyancy-induced nature of these instabilities. The most amplified frequencies from the linear stability analysis are in reasonable agreement with those determined from simulations of unstable triple flames. A parametric study using the linear stability analysis suggests that both the frequencies and amplitudes of disturbances increase with the magnitude of the gravity acceleration constant. Furthermore, the magnitude of amplification is largest just downstream of the two premixed branches. This trend implies the important role of baroclinic vorticity in the onset of instabilities. From the results of unsteady flame simulations, the onset of instability was found to correlate best with the Froude number based on premixed flame thickness and the triple flame propagation speed.