Microfluidic biochips enable low-cost automation of biochemical protocols with numerous applications to medical diagnostics, forensics, molecular biology, and drug design. An important component of protocol design is sample preparation, which involves dilution or mixing of two or more fluids in a desired ratio of concentration factors (CF). Existing continuous-flow microfluidic biochips deploy either free-flowing networks where only a single layer of flow-channels is used devoid of any control valves, or valve-based technology where the flow-layer is augmented with a control layer of valves. While the former is easy to fabricate, reliable, and less expensive, they are typically hardwired for specific applications only. The latter class, although programmable, is expensive and prone to various manufacturing and operational defects. In this paper, we present the physical design of a microfluidic network that is free-flowing as well as programmable. The proposed valve-free network resembles a complete binary tree with serpentine obstacles embedded within its channels, and can be used to achieve a desired dilution of a sample just by proper selection of fluid concentrations to be fed as inputs under constant pressure. Simulation with COMSOL Multiphysics Software shows that the proposed network provides a powerful and versatile architecture for solution preparation with minimal control, outperforming prior approaches in terms of the accuracy of CFs and time for convergence.