It appears I am expected to pay half of having a large tree cut down but it's not on my property. What should I do?

Between my house and the neighbor’s house is a large tree that is slightly on her side of the yard. The branches hang over my house; not hers. Then the leaves collect in my gutters. She hinted that I would pay half of having that tree cut down which would cost me about $450 ! She just moved in and is very nice . I dont want to be rude to her but it is not on my property and does not even touch her house ! If I call in the POA to decide she’ll probably dislike me forever. Homeownership is not easy.