Optimal Binary Trees with Order Constraints

Given a sequence of numbers ul,. ,a,,, find a binary tree with q leaves minimizing max{/l, + (II,. , II,, + u, }, where h, is the distance from the ith leaf to the root, i = I, , 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 y subtrees of heights (II,. , uy whose leaves partition the leaves of the tree. This question stems 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).