The relationship between the ‘shape of an organ’ and the ‘cost of blood transport’ to perfuse its tissue was evaluated on the basis of optimized arterial model trees simulated to perfuse square-based 100-cm<sup>3</sup> volumes of different shape (‘flat’ versus ‘thick’ as defined by the ratio of thickness to side-length h/s ≤1). Specifically, the effects of ‘shape’ on tree structure, blood transport, and on hemodynamic characteristics were investigated. Branching models of arterial trees were generated by constrained constructive optimization (CCO), based on an identical set of model parameters. All model trees were geometrically and topologically optimized for intravascular volume. Tree structures achieved tremendous savings of blood (transport medium) in comparison to a system of separate tubes. Thickening the perfusion volume (increasing h/s) resulted in a significant decrease of mean transport length, deposition time, and intravascular total volume in the tree. ‘Thick’ perfusion volumes induced CCO trees to branch more symmetrically into a number of equivalent subtrees repetitiously splitting into smaller ones; ‘flat’ structures were dominated throughout by a few asymmetrically branching major vessels. In summary, we conclude from systematic variation of shape that thicker perfusion volumes ( h/s >0.1) facilitate efficient delivery of blood in comparison to large amounts of ‘dead volume’ to be carried over long distances in very thin pieces of tissue.