Although much is known about the hydraulics of xylem, the hydraulic interconnectivity and dimensional scaling of phloem with respect to xylem in leaves has not been adequately studied to test alternative hydraulic architectural rules such as da Vinci's rule or Murray's rule, or physiological models such as Münch's Pressure Flow hypothesis. Using confocal and electron microscopy as well as mathematical analyses, we examined the hydraulic architecture of the mature leaves of the model species Populus tremula × alba across all seven hierarchical orders of the vascular branching. We show that: phloem and xylem conductive areas increase from minor to major veins; the sum of the conductive areas for each vein order increases exponentially from major to minor veins; the volume of individual sieve tube and vessel members increases from minor to major veins; and phloem conductive area scales isometrically with respect to xylem area across all vein orders. The application of first principles to our data shows that conductive areas scale according to da Vinci's rule and not according to Murray's rule, and that the phloem network in poplar leaves can generate the pressure gradient envisioned in Münch's hypothesis.