FUNCTIONAL PEARLS Drawing Trees

This article describes the application of functional programming techniques to a problem previously studied by imperative programmers, that of drawing general trees automatically. We first consider the nature of the problem and the ideas behind its solution (due to Radack), independent of programming language implementation. We then describe a Standard ML program which reflects the structure of the abstract solution much better than an imperative language implementation. We conclude with an informal discussion on the correctness of the implementation and some changes which improve the algorithm’s worst-case time complexity. 1 The problem and its solution The problem is this: given a labelled tree, assign to each node a position on the page to give an aesthetically pleasing rendering of the tree. We assume that nodes at the same depth are positioned on the same horizontal line on the page, so the problem reduces to finding a position horizontally for each node. But what do we mean by “aesthetically pleasing”? The various papers on the subject (Radack, 1988; Wetherell and Shannon, 1979; Vaucher, 1980; Reingold and Tilford, 1981; Walker, 1990) list aesthetic rules which constrain the positions in a number of ways. We adopt the same rules as Radack and Walker: 1. Two nodes at the same level should be placed at least a given distance apart. 2. A parent should be centred over its offspring. 3. Tree drawings should be symmetrical with respect to reflection—a tree and its mirror image should produce drawings that are reflections of each other. In particular, this means that symmetric trees will be rendered symmetrically. So, for example, Figure 1 shows two renderings, the first bad, the second good. 4. Identical subtrees should be rendered identically—their position in the larger tree should not affect their appearance. In Figure 2 the tree on the left fails the test, and the one on the right passes. Finally, trees should be as narrow as possible without violating these rules. 528 Andrew J. Kennedy