I don't think that a parse tree is intrinsic -- it's just something that maps well to the languages that we actually use: of course you could translate everything to a turing machine's tape, but that wouldn't be particularly useful.
What would really interest me would be exploring that if human logic fundamentally reduces to a parse tree because of something to do with the language faculty of the mind, and the programming languages that we create tend towards that just because programming languages are built out of the same logical faculty.
Yeah, that's the point. Other representations are possible (binary), but only one has proven suitable for programming. This is presumably because of the way the human mind works. It would be remarkable if it were the only good representation for this. It would also be the reason why Lisp is the limiting case among programming languages, at least when it comes to metaprogramming. But I repeat myself :)
What would really interest me would be exploring that if human logic fundamentally reduces to a parse tree because of something to do with the language faculty of the mind, and the programming languages that we create tend towards that just because programming languages are built out of the same logical faculty.