8)A simple experiment with a ruler and paper shows that any position for the gallows leads to the same point. However, it can be proven with a bit of vector algebra...
Define a square grid with its origin at tree A (herein point A) tree B at  (0,1). Gallows G exist at arbitrary point (x,y). The walk from G -> A == (-x,-y) The walk from G -> B == (-x,-y+1) Let A* denote the position after turning left at tree A, and B* denote the position reached after turning right at tree B. The walk from A -> A* == (+y,-x)
The walk from B -> B* == (-y+1,x) (not entirely obvious, but true!) The walk from G -> A* == (G -> A) + (A -> A*) == (-x+y , -x-y ) The walk from G -> B* == (G -> B) +
(B -> B*) == (-x-y+1 , x-y+1) The walk from A* -> B* == (G -> B*) - (G -> A*) == (-x-y+1+x-y , x-y+1+x+y ) == (1-2y , 1+2x) Now, the walk to the treasure is (A->A*) + .5(A*->B*) since the treasure is half way from A* to B* == (+y,-x) + .5(1-2y , 1+2x) == (+y+.5-y , -x+.5+x) == (.5,.5) So stand at A, walk half way to B, turn RIGHT and walk the same distance straight ahead. Get digging.....