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Laser-scanned point clouds of forests make it possible to extract valuable information for forest management. To consider single trees, a forest point cloud needs to be segmented into individual tree point clouds. Existing segmentation methods are usually based on hand-crafted algorithms, such as identifying trunks and growing trees from them, and face difficulties in dense forests with overlapping tree crowns. In this study, we propose TreeLearn, a deep learning-based approach for tree instance segmentation of forest point clouds. Unlike previous methods, TreeLearn is trained on already segmented point clouds in a data-driven manner, making it less reliant on predefined features and algorithms. Furthermore, TreeLearn is implemented as a fully automatic pipeline and does not rely on extensive hyperparameter tuning, which makes it easy to use. Additionally, we introduce a new manually segmented benchmark forest dataset containing 156 full trees, and 79 partial trees, that have been cleanly segmented by hand. The data is generated by mobile laser scanning and contributes to create a larger and more diverse data basis for model development and fine-grained instance segmentation evaluation. We trained TreeLearn on forest point clouds of 6665 trees, labeled using the Lidar360 software. An evaluation on the benchmark dataset shows that TreeLearn performs equally well or better than the algorithm used to generate its training data. Furthermore, the method's performance can be vastly improved by fine-tuning on the cleanly labeled benchmark dataset. The TreeLearn code is available from https://github.com/ecker-lab/TreeLearn. The data as well as trained models can be found at https://doi.org/10.25625/VPMPID.