Many organs are composed of tubular networks that arise by branching morphogenesis in which cells bud from an epithelium and organize into a tube. Fibroblast growth factors (FGFs) and other signalling molecules have been shown to guide branch budding and outgrowth, but it is not known how epithelial cells coordinate their movements and morphogenesis. Here we use genetic mosaic analysis in Drosophila melanogaster to show that there are two functionally distinct classes of cells in budding tracheal branches: cells at the tip that respond directly to Branchless FGF and lead branch outgrowth, and trailing cells that receive a secondary signal to follow the lead cells and form a tube. These roles are not pre-specified; rather, there is competition between cells such that those with the highest FGF receptor activity take the lead positions, whereas those with less FGF receptor activity assume subsidiary positions and form the branch stalk. Competition appears to involve Notch-mediated lateral inhibition that prevents extra cells from assuming the lead. There may also be cooperation between budding cells, because in a mosaic epithelium, cells that cannot respond to the chemoattractant, or respond only poorly, allow other cells in the epithelium to move ahead of them.