No, you have completely misunderstood the situation, if you were to place a small shed in the middle of the land, your potential claim for AP would just be for the footprint of that shed..........and it would fail.
A sketch or anonymised title plan would greatly help the understanding but in the absence of that you need to treat the whole area of land as the owner and block any access to the neighbour.
What sort of tree is it that the neighbour is climbing down?............if it's a self set ash or sycamore, then just fell it, if it's something different or productive it may need a rethink.