"We should tie the rope to the tree."
The McGraw Hill Handbook of English Grammar and Usage gives two tests for determining if a PP is adjectival or adverbial, neither of which stands up to much scrutiny (pages 39-41). The 'movement test' for an adverb PP is that if the PP can be moved to a different position in the sentence, it is an adverb PP that modifies the verb.
In "We should tie the rope to the tree", 'to the tree' is absolutely definitely (in normal usage) a modifier to 'tie'. But we couldn't say "To the tree we should tie the rope" or "We should to the tree tie the rope". So the movement test fails here.
The test for an adjective PP is that it, together with the whole noun phrase, can be replaced with a pronoun. In theory, 'the rope to the tree' could be replaced by 'it', but only if you take the sentence to be "We should tie [the rope which runs to the tree] [and then tie that other rope too]." And that is not the way any normal user would understand the sentence. So this test is unreliable too.
Is there any reliable test or tests for whether a PP modifies the noun or the verb, or do we have to rely upon sense?