  For any positive integer $h$, a graph $G=(V,E)$ is said to be $h$-magic if
there exists a labeling $l:E(G)\to \mathbb{Z}_h -\{0\} $ such that the induced
vertex set labeling $\ l^+ : V(G) \to \mathbb{Z}_h \ $ defined by $$ l^+
(v)=\sum_{uv \in E(G)} \ l(uv) $$ is a constant map. The integer-magic spectrum
of a graph $G$, denoted by $IM(G)$, is the set of all $h \in \mathbb{N} $ for
which $G$ is $h$-magic. So far, only the integer-magic spectra of trees of
diameter at most five have been determined. In this paper, we determine the
integer-magic spectra of trees of diameter six and higher.
