Mr. Branches is a dead tree that appears in episode 7. In what is possibly just Salad Fingers' imagination, Mr. Branches has a face and speaks in a voice of a child. It is possible that Mr. Branches is a deceased child of Salad Fingers, and that he is putting his or her former personality in an inanimate object and hallucinating that it could talk.
Mr. Branches appears again in episode 9, where it is sticking one of its branches through Salad Fingers' doorway. Salad Fingers bites off the end of the branch, prompting Mr. Branches to start crying, calling him "Daddy."
Mr. Branches seems to want to come inside Salad Fingers' home, however it must come inside by itself, as Salad Fingers won't help it. It is highly possible that Salad Fingers was hallucinating that he is Mr. Branches' father (it is unknown if Mr. Branches is inanimate or was once a human but was mutated and/or died.) In reality the tree cannot talk, move, or come inside because it is rooted to the ground.
There is a chance that due to Mr. Branches being capable of speaking English (combined with the other mutated individuals) that Mr. Branches could indeed be real.
It's also possible that it is a mutated tree that cannot talk but, indeed did intrude on the house. (Possibly to out of curiosity or attempting to eat Salad Fingers.) It also seems to be hostile towards him to some degree, as it violently constricted him with one of its roots. This may or may not have triggered Yvonne's coming in some way.