  Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges
$E(G)$. A subset $S$ of $E(G)$ is called a $k$-nearly independent edge subsets
if there are exactly $k$ pairs of elements of $S$ that share a common end.
$Z_k(G)$ is the number of such subsets. This paper studies $Z_1$. Various
properties of $Z_1$ are discussed. We characterise the two $n$-vertex trees
with smallest $Z_1$, as well as the one with largest value. A conjecture on the
$n$-vertex tree with second-largest $Z_1$ is proposed.
