  A genus one labeled circle tree is a tree with its vertices on a circle, such
that together they can be embedded in a surface of genus one, but not of genus
zero. We define an e-reduction process whereby a special type of subtree,
called an e-graph, is collapsed to an edge. We show that genus is invariant
under e-reduction. Our main result is a classification of genus one labeled
circle trees through e-reduction. Using this we prove a modified version of a
conjecture of David Hough, namely, that the number of genus one labeled circle
trees on $n$ vertices is divisible by $n$ or if it is not divisible by $n$ then
it is divisible by $n/2$. Moreover, we explicitly characterize when each of
these possibilities occur.
