A planar flow network consisting of successive generations of bifurcating vessels located downstream from a single mother vessel containing an incident fully developed flow is investigated. The theory and analysis developed which are for relatively thin vessels apply to small, medium or large networks. Although each successive bifurcation is in effect from a new mother vessel to two daughters, the networked system splits these into different types of bifurcation, the middle ones being inertial and the edge ones being viscous–inviscid in view of the wall conditions. The influences of network shapes, topology and end-pressure differences on the flow ahead of and inside the network are examined. Distinct local and global forms of upstream influence are active. The effects are especially marked in terms of non-symmetry, which leads to a global upstream influence, displaces the whole incident flow and particularly affects the motions near the outermost walls; there the non-symmetrical effects govern the induced wall shear stress and pressure and the solution dependence is very sensitive because of the realistic incident flow. Results from lattice-Boltzmann simulations are also described, and comparisons are then made with the theory and analysis. Pressure and shape control are considered in detail.