At runtime, the original program (i.e., the syntax tree, or some other representation from which the syntax tree is recoverable) may well not exist anymore. If you don't have a syntax tree, you can't type-check anything.
Now, you may say “okay, the original program doesn't exist anymore, but my language implementation is a term rewriting engine, so I have a current program that I can submit to the type checker”. Alas, most type checkers, Hindley-Milner included, operate on self-contained program fragments (i.e., not containing unbound variables), so you can't just submit any arbitrary program fragment. And, if you didn't want to type-check your original program, how likely is it that you will want to type-check the entire syntactic representation of the current program state?
Who says you need a syntax tree? You just store a byte of metadata next to each variable with the type. There should be more than enough data from that to recover the parts that you can't store like that and do type inference on them.
Now, you may say “okay, the original program doesn't exist anymore, but my language implementation is a term rewriting engine, so I have a current program that I can submit to the type checker”. Alas, most type checkers, Hindley-Milner included, operate on self-contained program fragments (i.e., not containing unbound variables), so you can't just submit any arbitrary program fragment. And, if you didn't want to type-check your original program, how likely is it that you will want to type-check the entire syntactic representation of the current program state?