All compilers translate a text stream into a tree structure during parsing, so arguing that some language "needs" infix is saying that there is some tree struct

All compilers translate a text stream into a tree structure during parsing, so arguing that some language "needs" infix is saying that there is some tree structure out there that can't be traversed depth-first rather than breadth-first. That can only be true if your supposed "tree" actually contains loops. What language do you know of that generates cyclical parse structures?