Are trees determined by their Laplace spectrum? I have to prove that a certain tree has a certain structure, and I've gotten as far as proving that it has the correct Laplace spectrum. While trees are definitely not determined by the spectrum of their adjacency matrix, I haven't been able to find anything on whether they're determined by their Laplace spectrum, with multiplicity (and if not, which ones are.) Is this a settled question, and if so where can I find a resource on it?

No. See http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees.pdf for an infinite family of counterexamples. (Published in Ars Combinatoria 3 (1977) 219-232.) I should point out that this paper proves that the proportion of trees on $n$ vertices that are determined by the characteristic polynomials of the Laplacian matrix, the adjacency matrix, and a number of other matrices (all together), goes to zero as $n\to\infty$. 
