<p>Stream networks express how Earth’s hydrologic cycle is embedded within its three-dimensional topography. In a top-down view, a stream network’s morphology is often described by its topological connectivity and branching geometry. Although these two characteristics are naturally connected, they have mostly been studied independently, leaving their co-evolution poorly understood. Here, we analyze the topology and geometry of 16,322 5^th-order real-world stream networks across the contiguous United States, showing how they are shaped by climate and the evolution of Earth’s topography. We find that ~73% of these networks show topological self-similarity in their branching patterns and that small tributaries join larger streams at systematically wider angles. Our analysis further reveals that correlations between climate and network topology observed in other studies are mainly mediated through the climate-dependence of networks’ geometric and topographic properties, such as their junction angles and channel slope ratios of merging tributaries. These findings demonstrate the co-evolution of network geometry, topography, and topology under the influence of landscape evolution driven by climatic forcing.</p>