Edge boundary of a finite forest in a regular tree
= Edge boundary of a finite forest in a regular tree
{title2=$|\partial_E A|=(k-2)|A|+2c(A)$}
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 $(k-2)|A|+2$.