We propose a model equation for the dynamics of tree density in mesic savannas. It considers long-range competition among trees and the effect of fire acting as a local facilitation mechanism. Despite short-range facilitation is taken to the local-range limit, the standard full spectrum of spatial structures obtained in general vegetation models is recovered. Long-range competition is thus the key ingredient for the development of patterns. The long time coexistence between trees and grass, and how fires affect the survival of trees as well as the maintenance of the patterns is studied. The influence of demographic noise is analyzed. The stochastic system, under the parameter constraints typical of mesic savannas, shows irregular patterns characteristics of realistic situations. The coexistence of trees and grass still remains at reasonable noise intensities.