[RWebb]

the laws also vary by state, and state law will control over city law, depending

typically, any part of their tree that crosses over the property line (term is encroachment) is "yours" and can be cut off - may look really ugly tho as you'd need to leave a stub over their property

Trees Standing on a Property Boundary are trickier, but looks like this one is clearly on his property with limbs (& roots) coming over onto your property

to avoid the high cost of an atty, find a book by Nola Press called "Neighbor Law"

BUT it won't protect you as well as hiring your own atty, if your neighbor gets ugly about it

you might talk to him & say, "we are going to fix that fence & may need to trim a few branches that come over onto our side"
"would you like us to cut them off at the trunk for you?"
or...
"hi how are ya? say, I wonder if you'd be interested in doing anything about that fence that's falling down in back there?"

I have nice neighbors (with one very bad exception); one of the good ones even let me have a guy come over and trim a branch that was completely on her property (blocking my view) - of course, I'm letting her use a 5 ft. strip of my yard for her garden too, so that helps.