Then, I don't understand your numbers. I'd say , , where corresponds to a number of balls of radius . (If the tree is in a square of unit sidelength). And in general hence you deduce . And .
Another reason why it is 1 is because if you cut the tree in halfs you almost have the same tree divided by a factor 2. It should be for dimension . (Think of integer values: you can cut a square in parts that look like the square to the scale ). I don't have a theorem however to justify this.