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The mechanisms regulating savanna tree populations are still not well understood. Recent empirical work suggests that both tree-tree competition and fire are key factors in semi-arid to mesic savannas. However, the potential for competition to structure savannas, particularly in interaction with fire, has received little theoretical attention. We develop a minimalistic and analytically tractable stochastic cellular automaton to study the individual and combined effects of competition and fire on savannas. We find that while competition often strongly depresses tree density, fire generally has little impact, but can drive tree extinction in extreme scenarios. When combined, competition and fire interact nonlinearly, magnifying each others negative effects on tree density a novel result that may help explain several observed phenomena in savannas. This strong interaction can also render competition difficult to detect in the field because fire can override the characteristic regular tree spacing driven by competition and lead instead to clustering.