Edge boundary of a finite forest in a regular tree ($|{\partial }_{E}A|=(k-2)|A|+2c(A)$)

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 $(k-2)|A|+2$.