Abstract
An algorithm of Knuth for finding an optimal binary tree is extended in several directions to solve related problems. The first case considered is restricting the depth of the tree by some predetermined integer K, and a $Kn^2 $ algorithm is given. Next, for trees of degree $\sigma $, rather than binary trees, $Kn^2 \log \sigma $ and $n^2 \log \sigma $ algorithms are found for the restricted and nonrestricted cases, respectively. For alphabetic trees with letters of unequal cost, $\sigma ^2 n^2 $ algorithm is proposed. We conclude with a comparison of alphabetic and nonalphabetic trees and their respective complexities.
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