The depth-first picture expression (DF-expression) is a compact quadtree expression providing a high data compression capability. It is primarily a hierarchically structured data representation for binary pictures. The authors first review the basic ideas of the DF-expression, and its potential data-compression competence. Then the practical problem is studied of how much reduction can be achieved data by converting the expression to a binary coding. It is also shown that three are two directions in the DF-expression strategy for the gray-image coding. One is the bit-plane based, and the other is the three-dimensional (three-valued) DF-expressions. For most of the natural images, the bit-plane based strategy is more effective than the three-dimensional one. The reason for that is discussed.