This is a Dependency syntax tree from an official website. As can be seen from this pic, "off" is deemed as a complete subtree. But when we use constituency tests to verify it, it does not perform as a constituent actually. I thought that "customer" was supposed to be a dependent of "off". Can somebody help me understand the place off "off" in this pic. Is it similar to the determiners in dependency trees?
This gets into the issue of phrasal verbs.
Sometimes in English, a verb connected to a preposition will act like, well, just a verb connected to a preposition. The preposition will dominate some other phrase, as prepositions normally do:
She lived in that apartment.
She lived, back in 2010, in that apartment.
It was in that apartment that she lived.
However, sometimes we see a similar structure where the preposition and the following phrase aren't a constituent together:
She handed in that assignment.
*She handed, back in 2010, in that assignment.
*It was in that assignment that she handed.
The second type are called phrasal verbs, and that's what "put off" is here:
Drunks could fall off the staircase.
Drunks could fall, if they're not careful, off the staircase.
Drunks could put off the customers.
*Drunks could put, if they're not careful, off the customers.
"Off" seems more closely bound to "put" than "the customers" here:
It's the customers that drunks could put off.
So it makes sense to say that "off" and "the customers" both depend on "put" (in different ways).
As a side note, this phrasal vs non-phrasal distinction can in some cases distinguish homophones in English:
He turned on his robotic ally. (could mean either "activated" or "betrayed")
He turned on, next, his robotic ally. (could only mean "activated")
He turned, next, on his robotic ally. (could only mean "betrayed")
It's one of the interesting quirks of English syntax.