The engineering and modeling of non-Newtonian power-law fluids (i.e., shear thinning and thickening fluids) in porous media has received wide attention in natural systems, oil recovery, and microfluidic devices. In this work, we theoretically explore the dynamics of power-law fluids in the form of capillary flow confined by a straight tube and a Y-shaped tree network, both of which are basic elements of many advanced materials. The straight tube and the tree network are composed of sub-tubes with different radii and lengths. The proposed model reveals that the evolution of the penetration time to the penetration distance is highly dependent on the viscous and capillary effects. If the viscous resistance is high, the flow is slow. If the capillary pressure increases, the flow accelerates. An interesting question is therefore in what optimal structure is the power-law flow fastest, considering different responses of viscous resistance and capillary force to the structural parameters. Based on optimization of the radius and length distribution of sub-tubes, we find the minimum penetration time of the fluids or the fastest flow in both straight tubes and tree networks under size constraints. The unique optimal transport behaviors of power-law fluids, which are different from those of Newtonian fluids, are analyzed in details with different power components. (C) 2015 Elsevier Ltd. All rights reserved.