We study inverse treatment planning approaches for stereotactic radiosurgery using Leksell Gamma Knife Perfexion (PFX, Elekta, Stockholm, Sweden) to treat brain cancer and tumour patients. PFX is a dedicated head-and-neck radiation delivery device that is commonly used in clinics. In a PFX treatment, the patient lies on a couch and the radiation beams are emitted from eight banks of radioactive sources around the patient's head that are focused at a single spot, called an isocentre. The radiation delivery in PFX follows a step-and-shoot manner, i.e., the couch is stationary while the radiation is delivered at an isocentre location, and only moves when no beam is being emitted. To find a set of well-positioned isocentres in tumour volumes, we explore fast geometry-based algorithms, including skeletonization and hybrid grassfire and sphere-packing approaches. For the selected set of isocentres, the optimal beam durations to deliver a high prescription dose to the tumour are later found using a penalty-based optimization model. We next extend our grassfire and sphere-packing isocentre selection method to treatments with homogenous dose distributions. Dose homogeneity is required in multi-session plans where a larger volume is treated to account for daily setup errors, and thus large overlaps with surrounding healthy tissue may exist. For multi-session plans, we explicitly consider the healthy tissue overlaps in our algorithms and strategically select many isocentres in adjacent volumes to avoid hotspots. There is also interest in treating patients with continuous couch motion to decrease the total treatment session and increase plan quality. We therefore investigate continuous dose delivery treatment plans for PFX. We present various path selection methods along which the dose is delivered using Hamiltonian paths techniques, and develop mixed-integer and linear approximation models to determine the configuration and duration of the radiation time along the paths. We consider several criteria in our optimization models, including machine speed constraints and movement accuracy, preference for single or multiple paths, and smoothness of movement. Our plans in all proposed approaches are tested on seven clinical cases and can meet or exceed clinical guidelines and usually outperform clinical treatments.