We propose an effective optimization algorithm for a general hierarchical segmentation model with geometric interactions between segments. Any given tree can specify a partial order over object labels defining a hierarchy. It is well-established that segment interactions, such as inclusion/exclusion and margin constraints, make the model significantly more discriminant. However, existing optimization methods do not allow full use of such models. Generic a-expansion results in weak local minima, while common binary multi-layered formulations lead to nonsubmodularity, complex high-order potentials, or polar domain unwrapping and shape biases. In practice, applying these methods to arbitrary trees does not work except for simple cases. Our main contribution is an optimization method for the Hierarchically-structured Interacting Segments (HINTS) model with arbitrary trees. Our Path-Moves algorithm is based on multi-label MRF formulation and can be seen as a combination of well-known a-expansion and Ishikawa techniques. We show state-of-the-art biomedical segmentation for many diverse examples of complex trees.