Are trees (connected acyclic graphs) determinable up to isomorphism by their spectra or characteristic polynomials? If not, what other pieces of information may help determine the tree?
Brendan McKay showed that there are many pairs of non-isomorphic trees with identical characteristic polynomials, as well as several other algebraic invariants. See Thm 4.2 in the linked paper.
Any tree is uniquely defined by the distances (the length of the shortest chain) between its leaves (vertices of degree 1); see:
Smolenskii Ye. A. A method for the linear recording of graphs Zh. Vychisl. Mat. Mat. Fiz., 2:2 (1962), pp.371–372.
K. A. Zaretskii, Constructing a tree on the basis of a set of distances between the hanging vertices”. Uspekhi Mat. Nauk, 20:6(126) (1965), pp. 90–92 (Russian).