Lateral gene transfer (LGT) is a common mechanism of non-vertical evolution where genetic material is transferred between two more or less distantly related organisms. It is particularly common in bacteria where it contributes to adaptive evolution with important medical implications. In evolutionary studies, LGT has been shown to create widespread discordance between gene trees as genomes become mosaics of gene histories. In particular, the Tree of Life has been questioned as an appropriate representation of bacterial evolutionary history. Nevertheless a common hypothesis is that prokaryotic evolution is primarily tree-like, but that the underlying trend is obscured by LGT. Extensive empirical work has sought to extract a common tree-like signal from conflicting gene trees. Here we give a probabilistic perspective on the problem of recovering the tree-like trend despite LGT. Under a model of randomly distributed LGT, we show that the species phylogeny can be reconstructed even in the presence of surprisingly many (almost linear number of) LGT events per gene tree. Our results, which are optimal up to logarithmic factors, are based on the analysis of a robust, computationally efficient reconstruction method and provides insight into the design of such methods. Finally we show that our results have implications for the discovery of highways of gene sharing.