What is the syntax of "second" in phrases like "the second most common problem"?

By | August 10, 2018

In English, words like "second", "third" etc. (also "next", I guess) can be used with a superlative to count down from the maximum.

Some dictionaries call "second" an adverb in this context (e.g. MW, AHD) but the OED covers this usage under its entry for the adjective.

It's not obvious to me how to analyze it. It does seem plausible to me that it is an adjective that modifies "most common problem" in a specialized way: (second (most common problem). This seems to be the structure given by the Link Grammar Parser by Davy Temperley, Daniel Sleator and John Lafferty:

    |              +---------------Ost---------------+     |
    |              |   +--------------Ds-------------+     |
    +---Wd---+-Ss*b+   +---L--+      +--EA--+----A---+     |
    |        |     |   |      |      |      |        |     |

LEFT-WALL this.p is.v the second.a most common.a problem.n . 

Constituent tree:

(S (NP This)
   (VP is
       (NP the
           (ADJP second)
           (ADJP (ADVP most)

But it also seems kind of like "second" is an adverb that modifies the meaning of "most common": ((second (most common) problem). Of course, "the second most common" can be used without any explicit following noun, but I don't know if this is relevant, since I guess that could be analyzed as a kind of ellipsis.

A third possibility that seems to feel right to some people (although I don't really get why) is "((second most) common) problem". I don't know if the fact that some people say things like "the most and second most important reason(s)" would support this analysis. I feel like the existence of phrases like "second oldest" or "second highest" counts as evidence against this analysis, because these examples show that "second" can be used with superlatives that don't contain the adverb "most".

Is there any published linguistic work that analyzes the syntax of this construction? If not, could someone post an answer that presents an analysis based on linguistic principles?