In this article we show that the global thermal resistance to flow between a volume and one point can be reduced to unprecedented levels by shaping the external boundary of each volume element. This degree of freedom is optimized, next to internal features such as the shape and volume fraction of the high-conductivity channels. The volume is covered in a sequence of optimization and assembly steps that proceeds toward larger sizes. The resulting architecture is a leaf-like tree structure with high-conductivity nerves and low-conductivity leaf material. The same constant resistance characterizes the flow from each point on the periphery of the structure to the common sink point. Nearly optimal structures in which the leaf shapes are replaced by needle-like (triangle-in-triangle) shapes are also developed. The fractal-like character of these designs and their relevance to the trend toward fractal-like properties in natural flow structures are discussed in the concluding section of the article. © 1999 American Institute of Physics.