Answer:
A.) The tree really needs to be cut down since it is old and half-rotted.
Explanation on why the others are incorrect:
B.) The tree really needs to be cut down; it is old and half-rotted: This is a coordinating conjunction (semicolon) used to join two independent clauses, which gives equal importance to both sentences, implying that the tree needs to be cut down and it is old and half-rotted are independent reasons.
C.) The tree really needs to be cut down, but it is old and half-rotted: This uses a coordinating conjunction (but) to join two independent clauses, but it implies a contrast or contradiction between the two statements, suggesting that the tree needs to be cut down despite it being old and half-rotted, which does not convey the intended meaning of the original sentences.