Could someone please tell me what's the dep. tree of "John seemed to fall asleep"? The problem is I don't know if "John" depends on "seem" or "fall".
Many modern DGs would give the sentence the following structure in surface syntax:
While seem is indeed a raising verb, that does not change the fact that the subject is generally positioned as an immediate dependent of the finite verb in many modern DGs. The position of John immediately in front of the seemed and the fact that John agrees with seemed (compare: *They seems...) are strong indicators that John is an immediate dependent of seemed.
If one chooses a DG approach that acknowledges both deep and surface syntax, then one could view John as a dependent of fall below the surface (i.e. in deep syntax).