On an Optimality Property of Ternary Trees

In the book of Knuth (1972) an extensive discussion can be found on some problems concerning rooted trees. Given a prescribed number of end-vertices, one may ask for the binary (or ternary) tree with minimal external path length (i.e. minimal sum of path lengths from the root to end-vertices), or minimal weighted external path length (here the end-vertices have given weights). In the first case the results are the complete binary. (or ternary) trees. In the second case the result is given by the Huffman algorithm. However, if one wants to construct an optimal tree, there is a problem prior to that of determining the binary tree with minimal external path length. I t is the problem of choosing the structure of the tree, given some criterion derived from the employment of the tree. One might want to deri~:e the structure of the tree instead of assuming the structure from the beginning. Several criteria may be considered. The one considered in this paper, in several variants, is the minimality of the "effort". Intuitively, this means the following. Consider the unique path from the root to an end-vertex. Each of the vertices of this path has a number of sons. Let the internal vertex v have u sons. The effort e to find the proper son will depend on u, and can reasonably be assumed to be an increasing function of u. Finding the proper end-vertex will require an effort equal to the sum of the efforts at the internal vertices of the path. The effort of the tree may then be defined as the sum of the efforts of all paths to end-vertices. However, it is clear that one might consider other criteria, e.g. the maximum (rather than the sum) of the efforts of all paths. In section 2 some preliminary results are derived for the case where the effort of an internal vertex is simply the number of sons. In section 3 it is shown that ternary trees require minimal effort for that effort function. The asymptotic