Tropical forests are a key component of the global carbon cycle. With plans for upcoming space-borne missions like BIOMASS to monitor forestry, several airborne missions, including TropiSAR and AfriSAR campaigns, have been successfully launched and experimented. Typical Synthetic Aperture Radar Tomography (TomoSAR) methods involve complex models with low accuracy and high computation costs. In recent years, deep learning methods have also gained attention in the TomoSAR framework, showing interesting performance. Recently, a solution based on a fully connected Tomographic Neural Network (TSNN) has demonstrated its effectiveness in accurately estimating forest and ground heights by exploiting the pixel-wise elements of the covariance matrix derived from TomoSAR data. This work instead goes beyond the pixel-wise approach to define a context-aware deep learning-based solution named CATSNet. A convolutional neural network is considered to leverage patch-based information and extract features from a neighborhood rather than focus on a single pixel. The training is conducted by considering TomoSAR data as the input and Light Detection and Ranging (LiDAR) values as the ground truth. The experimental results show striking advantages in both performance and generalization ability by leveraging context information within Multiple Baselines (MB) TomoSAR data across different polarimetric modalities, surpassing existing techniques.