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The Gamma-ray Large-Area Space Telescope (GLAST) balloon flight engineering model (BFEM) represents one of 16 towers that constitute the Large Area Telescope (LAT), a high-energy (> 20 MeV) gamma-ray pair-production telescope being built by an international partnership of astrophysicists and particle physicists for a satellite launch in 2006. The prototype tower consists of a Pb/Si pair-conversion tracker (TKR), a CsI hodoscopic calorimeter (CAL), an anticoincidence detector (ACD), and an autonomous data acquisition (DAQ) system. The self-triggering capabilities and performance of the detector elements have been previously characterized using positron, photon and hadron beams. External target scintillators were placed above the instrument to act as sources of hadronic showers. This paper provides a comprehensive description of the BFEM data-reduction process, from receipt of the flight data from telemetry through event reconstruction and background rejection cuts. The goals of the ground analysis presented here are to verify the functioning of the instrument and to validate the reconstruction software and the background-rejection scheme.