The adult vasculature is comprised of three distinct compartments: the arteries, which carry blood away from the heart and display a divergent flow pattern; the capillaries, where oxygen and nutrient delivery from blood to tissues, as well as metabolic waste removal, occurs; and the veins, which carry blood back to the heart and are characterized by a convergent flow pattern. These compartments are organized in series as regard to flow, which proceeds from the upstream arteries to the downstream veins through the capillaries. However, the spatial organization is more complex, as veins may often be found paralleling the arteries. The factors that control the morphogenesis of this hierarchically branched vascular network are not well characterized. Here, we explain how arteries exert a morphological control on the venous pattern. Indeed, during vertebrate development, the following transition may be observed in the spatial organization of the vascular system: veins first develop in series with the arteries, the arterial and venous territories being clearly distinct in space (cis-cis configuration). But after some time, new veins grow parallel to the existing arteries, and the arterial and venous territories become overlapped, with extensive and complex intercalation and interdigitation. Using physical arguments, backed up by experimental evidence (biological data from the literature and in situ optical and mechanical measurements of the chick embryo yolk-sac and midbrain developing vasculatures), we explain how such a transition is possible and why it may be expected with generality, as organisms grow. The origin of this transition lies in the remodeling of the capillary tissue in the vicinity of the growing arteries. This remodeling lays down a prepattern for further venous growth, parallel to the existing arterial pattern. Accounting for the influence of tissue growth, we show that this prepatterned path becomes favored as the body extends. As a consequence, a second flow route with veins paralleling the arteries (cis-trans configuration) emerges when the tissue extends. Between the cis-cis and cis-trans configurations, all configurations are in principle possible, and self-organization of the vessels contributes to determining their exact pattern. However, the global aspect depends on the size at which the growth stops and on the growth rate.