Is the library responsible for kudzu that is invading a neighbor's yard?

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Original Question: Sorry to ask what is probably a silly question on a Friday afternoon, but the owner of some property adjacent to the library has called to complain that our kudzu is overtaking her yard. Are we legally responsible for keeping her yard trimmed? Any advice would be most welcome.

Respondent: I cannot cite the GA Code for this so my response won't be definitive enough for Gary, but I can tell you from experience that FEMA regulations state that after a storm whatever is on your property is your problem -- be it the neighbor's roof, chicken coop, tree or kudzu.
I can tell you that I am responsible, according to the DNR, for cleaning up the neighbor's jetsam that washes up on my bottom land after a flood...and on my river bank anytime (even if the DNR catches the neighbor in the act of dumping it...)
I can tell you that we (the library) are responsible each fall to clean up the gazillion large sycamore leaves that blow on our yard, on our roof, blocking the gutters, from the neighbor's tree.
I can tell you that we (the library) have to pay the folks to come and cut down the wisteria growing from our neighbor's yard to keep it off our satellite dish, fence, trees, and whatever cars it has eaten in the last few hours...
Now, if your kudzu, uncut and unkempt, violates a local city ordinance for yard maintenance, well then it's your problem up to your property line.

Respondent: According to our city administrator, the kudzu in the neighbor’s yard is the neighbor’s problem, which confirmed the responses that I had received from all of you. I had checked our city code on Friday and could not find a reference to this situation, but I wanted to be sure before I confronted the woman. Thank you all for your help.
I seem to remove at least one healthy tree from my own yard each year because my 85-year-old neighbor is afraid they will fall on his property. Anything to keep the peace.

Respondent: It was wise that you did remove the tree. As Dusty pointed out, I like things regarding law to be definite. Though generally one is not responsible for a tree or things from a tree falling into a neighbor’s yard, there are exceptions. One of these is if the neighbor has pointed out that they consider a tree to be dangerous. Once that is brought to the owner’s attention, it then becomes the owner’s responsibility if damage does occur. So, if your neighbor said “That tree in your yard is leaning to one side and I’m afraid it may fall on my house” and then the tree did fall on her house, you would be responsible for damages. Of course, the neighbor would have to prove that you had been made aware, such as a copy of a letter or testimony from a third party.