Compounds are everywhere in English vocabulary, formed by combining two or more independent elements (‘free morphemes’, in linguistic jargon). They can be nouns (living room), verbs (download), adjectives (fun-loving), and other types. They can also be open, closed, or hyphenated, as shown.
The semantic relationship between the parts of a compound varies from one to another. Many are directly compositional; some require additional knowledge. When one element is part of the other, the main one tends to come first and be phonetically stressed: cliff edge, treetop, shoelaces, and so on.
So if we’re talking about the tip or tips of something, that’s the order we expect. Sure enough, there are fingertips, arrow tips, ear tips, horn tips, leaf tips, nerve tips, wingtips, and many more obscure compounds of the same structure. Which leads me to the present puzzle, which I aired first on Twitter: