Big tree in backyard. Good for rental and house value?

[Crystal Shi]

Hello everyone I just bought my first investment property in Southern California. It has a big shady tree in the backyard which looks nice. But it has extended over both my roof and the neighbor's. The leaves are falling crazily. Now I have to decide whether I should keep and trim it and remove it for the peace of mind. Some people say mature tree adds value to the house and some say it is hard to maintain. Animal may get in, root problem and roof damage. Any advice is appreciated. Thank you in advance.