Integral trees with given nullity

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique.
14 pages, 4 figures; This is a through revision of the first version including the correction of Lemma 13 (of first version) which was not correct as stated. We thank a referee for pointing out this mistake