By meaning not grammar. Compare with:
“I like the man in the building.”
What does “in the building” modify? Man not like. How do I know? The meaning makes more sense that way. If I wanted to say I like the man and I’m doing that liking in the building, I’d word it differently. Like “While I’m in the building, I like the man”.
A prepositional phrase in that position can modify the verb’s object.
But for “Then they hang it up in a tree.” if you meant “Then they hang up the one that is (already) in a tree” you’d word it differently to get that meaning across, like I just did. Or like, awkwardly, “Then they hang the in-a-tree it up”. Otherwise there’s no way to tell that “in a tree” is meant to tell us which “it” rather than where the hanging or up is.
Your intuition can answer a lot of questions like this for you. If it doesn’t tell you right away, you can try asking it more questions. Ask about related or similar sentences, simpler versions, versions where your intuition gives a different answer (like my man and building example), etc. Quizzing your intuition in a bunch of different ways can draw more information out of it and give you more confidence about what it says and what the boundaries are for when it will instead say something different.
Is there any ambiguity with ‘then’? Could ‘then’ be modifying the whole sentence? Is that an important distinction?
In general, I wouldn’t worry about “modifies the verb (that leads the clause)” vs. “modifies the clause”.