Yes, Brendan McKay showed that almost all trees have mates that are simultaneously cospectral in both adjacency and Laplacian spectra. And more.
http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees.pdf
Edit: I wondered briefly what the smallest pair of such trees would be, and a few minutes of Sage told me that there are two on 11 vertices.
And here they are (for some reason I am having difficulty with the image uploader):
