We investigate the fluid flow through two-dimensional ramified structures by direct simulation of the Navier-Stokes equations. We show that for trees with n generations, the flow distribution strongly depends on the Reynolds number Re. Specifically, for a tree without loops the flow becomes highly heterogeneous at high Re. For a tree with loops, on the other hand, the flow distribution tends to be more uniform at increased Re conditions. We show that these apparently contradictory behaviors have the same origin, namely, the effect of inertia on the momentum transport in the channels of the ramified geometry. In order to simulate the propagation of the flow imbalance throughout the tree without loops, we develop a simple model that incorporates the basic fluid dynamics features of the system. For large trees, the results of the model indicate that the distribution of flow at the outlet branches can be described by a self-affine landscape. Finally, we argue that the nonuniform partitioning of flow found for the structure without loops may contribute to the morphogenesis and functioning of the bronchial tree.