We generalize Schwenk's result that almost all trees contain any given limb to trees with positive integer vertex weights. The concept of characteristic polynomial is extended to such weighted trees and we prove that the proportion of $n$-vertex weighted trees that is weighted cospectral to another $n$-vertex weighted tree approaches $1$ as $n$ approaches infinity. We also prove that, for any integer $k\ge 2$, the proportion of $n$-vertex trees containing $k$ non-similar cospectral vertices approaches $1$ as $n$ approaches infinity.