{−1, 0, 1}-Basis for the null space of a forest

In this article we show that for any forest there exists a labelling of the vertices for which the row-reduced echelon form of its adjacency matrix is a {−1, 0, 1}-matrix. This result clearly provides an affirmative answer to the conjecture: The null space of the adjacency matrix of every forest has a {−1, 0, 1}-basis.