  A spectral faux tree with respect to a given matrix is a graph which is not a
tree but is cospectral with a tree for the given matrix. We consider the
existence of spectral faux trees for several matrices, with emphasis on
constructions.
  For the Laplacian matrix, there are no spectral faux trees. For the adjacency
matrix, almost all trees are cospectral with a faux tree. For the signless
Laplacian matrix, spectral faux trees can only exist when the number of
vertices is of the form $n=4k$. For the normalized adjacency, spectral faux
trees exist when the number of vertices $n\ge 4$, and we give an explicit
construction for a family whose size grows exponentially with $k$ for $n=\alpha
k+1$ where $\alpha$ is fixed.
