The discriminating ability of the hyper-Wiener index (the ability to discriminate between isomers) is discussed on a class of acyclic structures (trees) including the molecular graphs of alkanes. For trees with 4 to 26 vertices, numerical data on the classes of trees with coincident indices were obtained. For classes with maximal power, the diagrams of the corresponding trees are presented. Using the index for processing chemical structural data is discussed.