A note on optimal doubly-chained trees

This note concerns the recent paper by Hu [2] dealing with doubly-chained trees of the type introduced by Sussenguth [9]. In the second part of the paper, Hu deals with the problem of constructing an optimum weighted doubly-chained tree, under the standard assumption that no key is allowed to prefix another, as found, for example, in Patt [6]. He states that for the weights and elements of Figure 1, the optimum doubly-chained tree with all nodes reachable in less than five steps is the one shown in Figure 2. This tree is obtained from the optimal binary tree constructed in Hu and Tan [3], using the technique found in Knuth [5]. But the doubly-chained tree constructed by this method comes from an artificially restricted class of doubly-chained trees, and is thus not optimal in the sense usually defined in the literature [6-9] and described below. This fact is recognized by Hu [1], but was not explicitly stated in the paper in question.

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