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Aim To determine the functional relationships between, and the relative importance of, different driver variables (mean annual precipitation, soil properties, fire and herbivory) in regulating woody plant cover across broad environmental gradients in African savannas. Location Savanna grasslands of East, West and Southern Africa. Methods The dependence of woody cover on mean annual precipitation (MAP), soil properties (texture, nitrogen mineralization potential and total phosphorus), fire regimes, and herbivory (grazer, browser + mixed feeder, and elephant biomass) was determined for 161 savanna sites across Africa using stochastic gradient boosting, a refinement of the regression tree analysis technique. Results All variables were significant predictors of woody cover, collectively explaining 71% of the variance in our data set. However, their relative importance as regulators of woody cover varied. MAP was the most important predictor, followed by fire return periods, soil characteristics and herbivory regimes. Woody cover showed a strong positive dependence on MAP between 200 and 700 mm, but no dependence on MAP above this threshold when the effects of other predictors were accounted for. Fires served to reduce woody cover below rainfall-determined levels. Woody cover showed a complex, non-linear relationship with total soil phosphorus, and was negatively correlated with clay content. There was a strong negative dependence of woody cover on soil nitrogen (N) availability, suggesting that increased N-deposition may cause shifts in savannas towards more grassy states. Elephants, mixed feeders and browsers had negative effects on woody cover. Grazers, on the other hand, depressed woody cover at low biomass, but favoured woody vegetation when their biomass exceeded a certain threshold. Main conclusions Our results indicate complex and contrasting relationships between woody cover, rainfall, soil properties and disturbance regimes in savannas, and suggest that future environmental changes such as altered precipitation regimes, N-enrichment and elevated levels of CO2 are likely to have opposing, and potentially interacting, influences on the tree–grass balance in savannas. [ABSTRACT FROM AUTHOR]