This paper is a study of a highly decomposable algorithm useful for the parallel generation of a contour surface display. The core component of this algorithm is a two-dimensional contouring algorithm that operates on a single, 2 x 2 subgrid of a larger grid. A model for the operations used to generate the contour lines for a single subgrid is developed. The inadequacies of the currently published algorithms, with respect to contour line generation for a subgrid, are pointed out in a brief review of the available literature. A data structure, the contouring tree, is introduced as the basis of a new algorithm for generating the contour lines for the subgrid. The construction of the contouring tree, and the completeness of an algorithm based upon the contouring tree, within the constraints of the contouring model are shown.