Optimal material utilization in metal-matrix heat sink is investigated using constructal design (CD) in combination with fin theory to develop a constructal tree of optimally shaped convective fins. The structure is developed through systematic growth of constructs, consisting initially of a single convective fin enveloped in a convective medium. Increasingly complex convective fin structures are created and optimized at each level of complexity to determine optimal fin shapes, aspect ratios, and fin-base thickness ratios. One result of the optimized structures is a functional grading of porosity. The porosity increases as a function of distance from the heated surface in a manner ranging from linear to a power function of distance with exponent of about 2. The degree of nonlinearity in this distribution varies depending on the volume of the heat sink and average packing density and approaches a parabolic shape for large volume. For small volume, porosity approaches a linear function of distance. Thus, a parabolic (or least-material) fin shape at each construct level would not necessarily be optimal. Significant improvements in total heat transfer, up to 55% for the cases considered in this work, were observed when the fin shape is part of the optimization in a constructal tree of convective fins. The results of this work will lead to better understanding of the role played by the porosity distribution in a metal-matrix heat sink and may be applied to the analysis, optimization, and design of more effective heat sinks for electronics cooling and related areas.