Abstract
Given a sequence of numbers a l , …, a q , find a binary tree with q leaves minimizing max h l + a l , …, h q + a q , where h i is the distance from the ith leaf to the root, i = l, …, q. This problem is solved by means of a O( q) algorithm and a tight upper bound for the minimum is given by an explicit formula. The task is equivalent to finding a binary tree of minimum height having q subtrees of heights a l , …, a q whose leaves partition the leaves of the tree. This question seems to be of general interest. In particular, it arises in the problem of the optimal decomposition of a tree into chains (Waksman, Tech. Report FC 95-06, August 1995).
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