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Prior work has shown that the order in which different components of the face are learned using a sequential learner can play an important role in the performance of facial expression recognition systems. We propose FaceTopoNet, an end-to-end deep model for facial expression recognition, which is capable of learning an effective tree topology of the face. Our model then traverses the learned tree to generate a sequence, which is then used to form an embedding to feed a sequential learner. The devised model adopts one stream for learning structure and one stream for learning texture. The structure stream focuses on the positions of the facial landmarks, while the main focus of the texture stream is on the patches around the landmarks to learn textural information. We then fuse the outputs of the two streams by utilizing an effective attention-based fusion strategy. We perform extensive experiments on four large-scale in-the-wild facial expression datasets - namely AffectNet, FER2013, ExpW, and RAF-DB - and one lab-controlled dataset (CK+) to evaluate our approach. FaceTopoNet achieves state-of-the-art performance on three of the five datasets and obtains competitive results on the other two datasets. We also perform rigorous ablation and sensitivity experiments to evaluate the impact of different components and parameters in our model. Lastly, we perform robustness experiments and demonstrate that FaceTopoNet is more robust against occlusions in comparison to other leading methods in the area.