The structure of humpback whale (Megaptera novaeangliae) songs was examined using information theory techniques. The song is an ordered sequence of individual sound elements separated by gaps of silence. Song samples were converted into sequences of discrete symbols by both human and automated classifiers. This paper analyzes the song structure in these symbol sequences using information entropy estimators and autocorrelation estimators. Both parametric and nonparametric entropy estimators are applied to the symbol sequences representing the songs. The results provide quantitative evidence consistent with the hierarchical structure proposed for these songs by Payne and McVay [Science 173, 587-597 (1971)]. Specifically, this analysis demonstrates that: (1) There is a strong structural constraint, or syntax, in the generation of the songs, and (2) the structural constraints exhibit periodicities with periods of 6-8 and 180-400 units. This implies that no empirical Markov model is capable of representing the songs' structure. The results are robust to the choice of either human or automated song-to-symbol classifiers. In addition, the entropy estimates indicate that the maximum amount of information that could be communicated by the sequence of sounds made is less than 1 bit per second.