What have you all done in the past about neighbors trees overhanging your property/house/shed etc?

I have this problem on a house I recently purchased - about to start rehabbing. Now the tree hung down on the back of the roof and siding - so some of this will have to be replaced/fixed - just a little wear and tear nothing major. However, the tree comes up to the top of the ffence line the abruptly turns toward my property and partially over the house. The part hanging on the house has been trimmed by myself. 

From a legal standpoint - it looks like I can cut everything on my side - which is essentially the entire tree (however the truck is on her side of the fence). The downside is that this obviously will cost me - can I afford it - yes. But it's not my effing tree. Looks like I can also file a nuisance claim in court since the tree will eventually fall on the house and prevents "enjoyment" of my property. It would be obvious to a reasonable person that the tree needs to be trimmed. 

Do I want to let it eventually fall on the house - no. Don't need to have a tenant smashed or pay a 5k deductible. 

From a neighborly perspective - I can offer to split the cost with her to a degree. 

What have you all done it the past in situations like this? (It seems it can differ wildly by location)