The tree algorithm described in this monograph is an algorithm which maximizes functions of systems of linear relations subject to constraints. Typical problems in this class are concerned with identifying all of those vectors which satisfy or don4 satisfy given linear equalities or inequalities in such patterns as will maximize certain functions of interest. For example, consider the problem of identifying all of those vectors which satisfy as many of an inconsistent system of linear inequalities as possible. For another example, consider two overlapping multidimensional clouds of O's and x's; in this setting, the problem is to determine all quadratic hypersurfaces which best separate the clouds in the sense of having the fewest number of O's on the x side of the surface and vice-versa. Also, as very special cases, this class includes the problems of solving linear programs and systems of linear equations.