Forest proof of trifurcation boundary counting

Delete a finite set of degree-three trifurcation vertices in percolation in each infinite percolation cluster and contract the remaining components to nodes. The incidence graph is a tree: a cycle would bypass an original trifurcation vertex in percolation. All leaf components are infinite, otherwise a graph vertex would have a finite branch after deletion. The tree degree identity gives at least two more infinite leaves than degree-three graph vertices. Distinct infinite components escape a containing box through distinct outer boundary graph vertices. Hence the number of counted trifurcations is at most the outer boundary size.