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The Horton-Strahler number naturally arose from problems in various fields, e.g., geology, molecular biology and computer science. Consequently, detailed investigations of related parameters for different classes of binary tree structures are of interest. This paper shows one possibility of how to perform a mathematical analysis for parameters related to the Horton-Strahler number in a unified way such that only a single analysis is needed to obtain results for many different classes of trees. The method is explained by the examples of the expected Horton-Strahler number and the related rth moments, the average number of critical nodes, and the expected distance between critical nodes.