What is the ruling on eating from a tree that our neighbor, the owner, permitted us to eat from before her death? And is it necessary to inform her relatives living in her house of this permission?

The permission of the deceased ends with her death, and the right transfers to her heirs. You are not allowed to take from the tree unless the rights-holders (the heirs) permit you, either explicitly or implicitly through custom, for permission by custom is like explicit permission.