The complexity of branching drainage networks varies significantly among landscapes, spanning a spectrum from simple, straight channels that flow down the regional steepest gradient to complex, highly tortuous flow patterns. The source of the variability in complexity is puzzling because models show that this variability can develop even in the absence of environmental heterogeneity. Here, we propose quantifying network complexity by examining the distribution of lengthwise asymmetry between pairs of flow paths diverging from divides and rejoining at junctions. Using this measure of asymmetry, we demonstrate that the channel concavity index, which characterizes how channel slope changes downstream, exerts primary control over the complexity of natural drainage networks. Analytical modeling and simulations of optimal channel networks guided by an energy-minimization principle indicate that landscapes with low-concavity index channels maintain stability only through simplified network geometries. Conversely, landscapes characterized by high channel concavity index can sustain stable configurations with various degrees of complexity, including extremely intricate networks. As a result, landscapes featuring high-concavity channels tend to retain the imprint of former environmental conditions, while landscapes with low-concavity channels frequently reorganize in response to environmental shifts, effectively overwriting past patterns. Consistent with previous findings showing that channel concavity index correlates with climate aridity, we find a significant empirical correlation between aridity and network complexity, suggesting a climatic signature embedded in the large-scale planform geometry of landscapes.