Yes, it can spread to a tile with 3 branches as well (although when my character got burnt I was actually cutting branches, so there was a pile at the feet, but spontaneous spreading to 3 branch tiles is a thing as well).
It can be noted that as far as I've seen, it's safe to light a fire under the feet of a dog, as it will move out of the way (sometimes it moves onto the tile as you're trying to set it on fire, but that's safe as well).