The connection between monophyly and efficient taxonomic diagnoses is elaborated. The inefficiency of nonmonophyletic groups is shown by reconstructing data matrices from hierarchical sets of diagnoses that are derived from apomorphies and read in order from highest to lowest rank. The practice of diagnosing nonmonophyletic groups either results in omitting data, resulting in errors in reconstructed datasets, or repeating character information to make up for the implied losses. Step-by-step demonstrations with hypothetical and real data are used as guidance. Provisions are made for missing, inapplicable and polymorphic data. Slow optimization (delayed transformation) is useful for choosing a state reconstruction in order to report apomorphies completely. The diagnoses of paraphyletic groups can be expressed in different ways, including regrafting derived clades, reanalyzing data with constraints, and reading the original diagnoses in a different order--the last is the least efficient. A cladistic version of the data compression ratio is proposed to quantify the diagnostic efficiency of a cladogram.