Information on forest composition, and specifically the tree types and their distribution, aids in timber stock calculation and helps better understand the biodiversity in a particular region. Automatic satellite imagery analysis can significantly accelerate the process of tree type classification, which is traditionally done via ground-based observations. Although computer vision methods have proven their efficiency in remote sensing tasks, specific challenges arise in forestry applications. In this paper, we aimed to improve tree species classification based on the neural network approach. The study involved four species commonly found in the Russian boreal forests: birch, aspen, pine, and spruce. We used imagery from the Sentinel-2 satellite, which has multiple bands in the visible and infrared spectra, and spatial resolution of up to 10 meters. Additionally, the short revisit time and free access policy, it makes these images a valuable data source for the purposes of forest classification. In computer vision terms, we define the problem of tree type classification as one of semantic segmentation, and assign each pixel of the image a particular tree type. The forest inventory data contain the tree type composition, but do not describe their spatial distribution within each individual stand. Therefore, some pixels can be assigned a wrong label if we consider each stand to be homogeneously populated by its dominant species. This calls for the weakly supervised learning approach. To solve this problem, we used a deep convolutional neural network with a tailored loss function set to optimize for multiple objectives. We tested the model's transferability by creating a dataset of images for three regions of Russia (Arkhangelsk Oblast, Leningrad Oblast, and Perm Krai), with the combined area of more than one million hectares, over four years (2016-2019). In our study, we demonstrated how to modify the training strategy so it outperforms the basic per-pixel neural network approach.