In Mathematical Morphology, the max-tree is a region-based representation that encodes the inclusion relationship of the threshold sets of an image. This tree has proved useful in numerous image processing applications. For the last decade, work has led to improving the construction time of this structure; mixing algorithmic optimizations, parallel and distributed computing. Nevertheless, there is still no algorithm that benefits from the computing power of the massively parallel architectures. In this work, we propose the first GPU algorithm to compute the max-tree. The proposed approach leads to significant speed-ups, and is up to one order of magnitude faster than the current State-of-the-Art parallel CPU algorithms. This work paves the way for a max-tree integration in image processing GPU pipelines and real-time image processing based on Mathematical Morphology. It is also a foundation for porting other image representations from Mathematical Morphology on GPUs.