Forest

A forest is an undirected graph with no graph cycle; each of its connected components of a graph is a tree, including any isolated vertex. A finite forest with $n$ vertices and $c$ connected components of a graph hasIndeed, a connected component of a graph with $n_i$ vertices contributes $n_i-1$ edges; adding these counts gives $n-c$. Equivalently, start with $n$ isolated vertices: every added edge must join different connected components of a graph, reducing their number by one. Joining vertices already connected would create a graph cycle. ^[ref]

Deleting $13$ in our example gives $n=7$, $c=2$, and five edges: the two trees contribute $3+2=5=7-2$. Deleting $67$ as well leaves $7$ isolated, so $c=3$ and four edges. A forest therefore describes separate hierarchies or the intermediate stages of building a spanning tree. The acyclic graph condition accounts for missing connections without assuming that all locations are mutually reachable.

A two-component spanning forest of a finite connected graph is an acyclic spanning subgraph with exactly two connected components of a graph. It has $|V|-2$ edges. For distinct terminals $a,z$, a separating two-component forest places them in different components. Deleting an edge of the $a$-to-$z$ graph path of a spanning tree produces such a forest; adjoining any original edge across its cut reverses this operation. This is the counting correspondence behind the mean spanning-tree path current.