You can remove up to a third of a tree w/o causing damage. I don't think it;s a honey locust though, maybe a black locust. Honey locust have thorns all over the trunk. I wouldn't cut it mysef if it were hanging over the house, but a trained arborist would have no problem. My best buddy is an arborist. Probably cost you a couple hundred bucks to get that taken off there. If there are powerlines anywhere around, you may be able to get the power company to come take it off for you, but I don't see any in the pic.
To RR, you are supposed to cut limbs close to the collar of the tree, the thick part where the limb comes out, that supports the healing process. If a limb is left too long, that's when the rot happens.
Good luck, dpg