We present a tree-search algorithm for two-dimensional cutting problems in which there is a constraint on the maximum number of each type of piece that is to be produced. The algorithm limits the size of the tree search by deriving and imposing necessary conditions for the cutting pattern to be optimal. A dynamic programming procedure for the solution of the unconstrained problem and a node evaluation method based on a transportation routine are used to produce upper bounds during the search. The computational performance of the algorithm is illustrated by tests performed on a large number of randomly generated problems with constraints of varying “tightness.” The results indicate that the algorithm is an effective procedure for solving cutting problems of medium size.