It’s a widely-reported fact* that Dr. Seuss selected the words in The Cat in the Hat from a list of 348 words given to him by William Spaulding, the director of the education division of the publishing company, Houghton Mifflin.
Number of words on the original list given to Dr. Seuss: 348
Number of unique words he was supposed to use: 225
Number of unique words he actually used: 236
Total number of words in the book: 1,629
Words used only once: 54
Words used just twice: 33
Number of one-syllable words: 221
Number of two-syllable words: 14
Number of three-syllable words: Only 1--another
For fun: here’s a link to an activity sheet that shows all 236 words in The Cat in the Hat. You c-o-u-l-d try to make a story from them, but I must warn you that it took the very talented Dr. Seuss at least nine months to do it, including (according to one of his tongue-in-cheek “true stories”) several months working in his Uncle George’s coal mine.
*But was there really a 348-word list? I have not been able to track it down, nor do the words in the book match popular word lists used at that time. I think there was a list, but that Dr. Seuss adapted it to include words that rhymed with words on it, along with compound words like plaything. The only person's name in the book—Sally—was the name of the little sister in the Dick and Jane readers.