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Laser-scanned point clouds of real-world objects are frequently reconstructed into meshes. While man-made objects often include large regular areas and symmetries, noisy data with high-frequency information, such as point clouds of vegetation, still pose an open problem. We introduce TreeStructor, a novel approach for isolating and reconstructing forest trees. The key novelty is a deep neural model that uses neural ranking to assign pre-generated connectable 3D geometries to a point cloud. TreeStructor is trained on a large set of synthetic generated point clouds. The input to our method is a forest point cloud that we first decompose into point clouds that approximately represent trees (TPC) and then into point clouds that represent their parts (PPC). We use a point cloud encoder-decoder to compute embedding vectors that retrieve the best-fitting surface mesh for each PPC from a large set of predefined branch parts. Finally, the retrieved meshes are connected and oriented to obtain individual surface meshes of all trees represented by the FPC. We qualitatively and quantitatively validate that our method can reconstruct forest trees with unprecedented accuracy and visual fidelity. TreeStructor outperforms the state-of-the-art reconstruction method for around 6% on quantitative metrics and 12% less error compared with QSM on low-quality scanned data.