What is unique about savannas that allows the coexistence of trees and grasses? What are the factors that govern that coexistence? Despite many scientific efforts during the last decades, these questions are still open. Recent empirical work suggests that tree-tree competition and external forcing like fire and varying weather conditions are key factors that regulate savanna dynamics. To tackle the combined effects of these mechanisms from a Statistical Physics point of view, we develop a minimalistic and analytically tractable stochastic model, which is a variant of the Contact Process. The model exhibits a transition from grassland to tree-grass coexistence, whose nature depends on the values of the internal parameters. We also study the spatial structure of tree distribution and characterize the mechanisms leading to clustering.