1) Take a body of text and convert all references of "her" to either "him" or "his". Basically, this question from SO, which seems like a neat project.
2) I'm assuming that you've studied the transformational grammar approach to trees (X-bar theory, etc.) There is this whole other syntactic theory called Head-Driven Phrase Structure Grammar (HPSG) which posits not the use of "hierarchy" for describing syntax, but uses attribute-value matrices as well.
One of the advantages of using this system is that you can describe syntax as directed acyclic graphs, rather than trees. Your prof might be able to point you to more/better resources, but this page (vaguely) explains the idea. Unfortunately, my experience has been that HPSG resources on the net are pretty lacking.
But, once you have a feel for HPSG feature structures, you can search the literature for people (linguists and computational linguists) who've come up with ways to use the graph representations for interesting things.
Maybe you could create a transformational tree-to-HPSG graph converter? (You'd have to know how things like "raising", "control", "passivizations", etc would be transformed into HPSG's "SLASH", "REL", etc.)