(a) He has one plot of vacant land at the back of his house.
A plot is a legally-owned piece of land. It may include a house and surrounding yard, or simply be land with no house. Your sentence suggests that his house is on one plot and there is another plot behind that one. If that's what you mean,
Not to split hairs, but this may be a regional thing. I can have a shed, a pool, a patio, a firepit, and several trees in my backyard. (I don't - I have only some trees), but to me, the backyard doesn't have any requirement for it to be vacant of anything except my main house.
I guess 'vacant' can have different connotations. I think of it as "no permanent structures like a house or factory".
Generally speaking, I'd say my backyard is empty, although it has a patio, a deck, etc. I wouldn't say that land is not empty, or not vacant, just because it has trees on it.
I wouldn't really say my backyard is vacant, because it is only part of
Ah, I see. I was thinking of "vacant" as "no structures." And I believe you were thinking of it in the sense of "the owner isn't claiming it or doing anything with it." Both make sense to me, now that I have this perspective.
I would feel okay about planting a garden or annual flowers in a vacant lot behind my house because if someone did come and buy it, I would only lose that yea