Step 1: Trace the blood flow. Venous blood from the gut, spleen, pancreas and gall bladder collects into the hepatic portal vein, which is the functional afferent vessel of the liver. On cross-sectional imaging it is the structure that reaches the porta hepatis and splits into right and left limbs running into the lobes.
Step 2: The liver is parcelled into Couinaud segments according to portal and hepatic-vein anatomy. Because each segment is fed by a portal branch, the imaged branching vessel must be the portal vein supplying those segments.
Step 3: Eliminate the others. The superior vena cava is a thoracic vessel returning to the right atrium and has no hepatic branches. The inferior vena cava lies posterior to the liver and collects the hepatic veins rather than dividing within the parenchyma. The splenic vein is merely a contributor that unites with the superior mesenteric vein to create the portal vein.
Step 4: The caudate lobe, segment I, has its own twin portal supply and drains independently to the IVC, underscoring that the branching inflow is portal. Hence the branching structure is the portal vein.\[\boxed{\text{Portal vein}}\]