  A topological index reflects the physical, chemical and structural properties
of a molecule, and its study has an important role in molecular topology,
chemical graph theory and mathematical chemistry. It is a natural problem to
characterize non-isomorphic graphs with the same topological index value. By
introducing a relation on trees with respect to edge division vectors, denoted
by $\langle\mathcal{T}_n, \preceq \rangle$, in this paper we give some results
for the relation order in $\langle\mathcal{T}_n, \preceq \rangle$, it allows us
to compare the size of the topological index value without relying on the
specific forms of them, and naturally we can determine which trees have the
same topological index value. Based on these results we characterize some
classes of trees that are uniquely determined by their edge division vectors
and construct infinite classes of non-isomorphic trees with the same
topological index value, particularly such trees of order no more than $10$ are
completely determined.
