Neurons are connected by complex branching processes that collectively process information for organisms to respond to their environment. Classifying neurons according to differences in structure or function is a fundamental piece of neuroscience. Here, by constructing new biophysical theory and testing against our empirical measures of branching structure, we establish a correspondence between neuron structure and function as mediated by principles such as time or power minimization for information processing as well as spatial constraints for forming connections. Based on these principles, we use Lagrange multipliers to predict scaling ratios for axon and dendrite sizes across branching levels. We test our predictions for radius scale factors against those extracted from images, measured for species that range from insects to whales. Notably, our findings reveal that the branching of axons and peripheral nervous system neurons is mainly determined by time minimization, while dendritic branching is mainly determined by power minimization. Further comparison of different dendritic cell types reveals that Purkinje cell dendrite branching is constrained by material costs while motoneuron dendrite branching is constrained by conduction time delay. We extend this model to incorporate asymmetric branching, where there are multiple different paths from the soma to the synapses and thus multiple interpretations of conduction time delay; one considers the optimal path and the other considers the sum of all possible paths, leading to different predictions. We find that the data for motoneurons show a distinction between the asymmetric and symmetric branching junctions, corresponding to predictions using different interpretations of the time-delay constraint. Moreover, the more asymmetric branching junctions are localized near the synapses, indicating that different functional principles affect the structure at different regions of the cell. Finally, we use machine-learning methods to classify cell types using functionally relevant structural parameters derived from our model. Incorporating branching level as a feature in classification in addition to parameters related to information flow improves performance across methods, suggesting that information flow drives localized differences in morphology. Future directions of this work include estimating specific parameters related to functional tradeoffs and myelination using numerical optimization and analyzing changes across stages of development.