Given a number $n$ of vertices,
a lower bound $d$ on the diameter, and a capacity function
$\Delta(k)\geq 2$, $k=0,1,\ldots,\lfloor n/2\rfloor$,
we consider the problem of
enumerating all unrooted trees $T$
with exactly $n$ vertices and a diameter at least $d$
such that the degree of each vertex
with distance $k$ from the center of $T$
is at most $\Delta(k)$.
We give an algorithm that generates
all such trees without duplication in $O(1)$-time delay
per output in the worst case using $O(n)$ space.
For example, our result implies that all alkanes,
structural isomers of chemical graph ${\tt C}_n{\tt H}_{2n+2}$
can be generated in $O(1)$-time delay without duplication.