Objectives. Neuron morphology critically defines cellular identity and function, informing cell typing, soma localization, and neurological disorder diagnosis. However, two main challenges hinder progress: the difficulty of learning meaningful representations from complex, tree-like structures, and the limited availability of expert annotations at scale. Approach. To address these challenges, we propose MorphSys, a self-supervised contrastive learning framework that complements a branch-aware module and a graph neural network (GNN)-based module. We present a branch-level representation of neuron morphology by introducing an inter-branch attention, which captures inter-branch relationships among dendrite branches that are overlooked by conventional graph models relying on node-level message passing. In parallel, the GNN-based module robustly encodes local morphological patterns across different GNN architectures, providing complementary fine-grained structural features. Main results. Comprehensive experiments on five datasets demonstrate the superior performance of MorphSys in downstream tasks. In self-supervised neuron morphological classification, MorphSys achieves a k-nearest neighbor accuracy (KNN-Acc) of 83.52% on the N7 dataset, outperforming the previous state-of-the-art by 3.77%. For soma location prediction, MorphSys attains the highest KNN-Acc of 83.99%, 67.32%, and 75.33% on the Brain Image Library, ACT, and Janelia MouseLight datasets compared to baseline methods. The repeated trials and statistical analysis also confirm that the observed performance gains are highly significant. Significance. These results highlight that MorphSys serves as an effective tool for learning robust representations of neuron morphology and performing morphology-driven neuronal analysis. The code is available at https://github.com/YuuYuuYuuY/MorphSys.