Black branch isn't sold, but black lumber, which you get from chopping the black branch, is. lt's 1250G or 875G per piece, depending on your chosen mode, & sold at Log House. I got all of my black branches from trading items to the plinth on the island. I have 99 saved up from that. I frequented the island to save up on materials such as the 100 wine required for rank unlock. Very helpful!