The number of 1-nearly independent edge subsets

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. Zₖ(G) is the number of such subsets. This paper studies Z₁. Various properties of Z₁ are discussed. We characterise the two n-vertex trees with smallest Z₁, as well as the one with largest value. A conjecture on the n-vertex tree with second-largest Z₁ is proposed.