Regular tree ($\mathrm{deg}(x)=k(x\in V)$)

An infinite connected tree is regular of degree $k$ if every vertex has exactly $k$ neighbours. Necessarily $k\ge 2$. For $k=2$ it is a two-sided infinite path. For $k\ge 3$, rooting it gives $k$ children at the root and $k-1$ children at each other vertex; this constructs the unique such tree up to graph isomorphism.

For a nonempty finite vertex set $A$ in a degree-$k$ regular tree, its induced forest has $|A|-c(A)$ edges, where $c(A)$ counts its components. Summing degrees and subtracting twice the internal edge count gives the displayed number of edges with exactly one endpoint in $A$. In particular it is at least $(k-2)|A|+2$.