The first sentence you’re asked to draw on HW7 contains a DP her brother’s Coke. This is an extension of the possession constructions we’ve looked at. What’s special about this one is that it has two possession relationships (her-brother, and brother-Coke). The trick here is just that any DP can be a possessor within another DP, that’s what makes these “recursive.” An easy way to approach this would probably be to sketch out the DP for her brother and for his Coke, and then replace the DP his in his Coke with the DP you sketched for her brother. They’re both legitimate DPs, either his or her brother are perfectly fine possessor DPs.

The other thing I noticed has to do with the capitalization of Coke. I meant, of course, the carbonated beverage distributed by the Coca-Cola company. However, note that this is not a proper name. Possession constructions are incompatible with definite DPs like proper names (“*the his dog”, “*his the dog”). If you put a proper name in that place, you can mean something, but it turns the meaning into something like possession of an instance of a kind. That is, “his James Bond” takes on some kind of meaning other than the (fictional) person James Bond possessed by him, it is something more like “his interpretation of the character” or “his action figure” or whatever, depending on the context. Something of which there are several, one of which can be possessed by him. It might have been more perspicuous if I hadn’t used something with a brand name like Coke; I could have de-capitalized it to make it clearer, except that those not hailing from the southern US might have then taken the sentence to mean something somewhat different from what I’d intended.

The reason the proper name thing is relevant is just that the head D of the topmost DP in her brother’s Coke should be the null ∅_GEN, and not ∅_PROPER.