Would for example "their" be divided into they and 's under the DP theory when writing out a tree?

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Yes, you could do that. Distributed Morphology (Halle & Marantz 1993) is a morphology framework in which you would have an abstract "they" and "'s" (in your case) which would finally be realized as one morpheme, "their".

Halle, M., & Marantz, A. (1993). Distributed morphology and the pieces of inflection. In K. Hale, & S. J. Keyser (Eds.), The view from building 20 (pp. 111-176). The MIT Press.

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