  Given a tree $T$, let $q(T)$ be the minimum number of distinct eigenvalues in
a symmetric matrix whose underlying graph is $T$. It is well known that
$q(T)\geq d(T)+1$, where $d(T)$ is the diameter of $T$, and a tree $T$ is said
to be diminimal if $q(T)=d(T)+1$. In this paper, we present families of
diminimal trees of any fixed diameter. Our proof is constructive, allowing us
to compute, for any diminimal tree $T$ of diameter $d$ in these families, a
symmetric matrix $M$ with underlying graph $T$ whose spectrum has exactly $d+1$
distinct eigenvalues.
