  In the game of pegging, each vertex of a graph is considered a hole into
which a peg can be placed. A pegging move is performed by jumping one peg over
another peg, and then removing the peg that has been jumped over from the
graph. We define the pegging number as the smallest number of pegs needed to
reach all the vertices in a graph no matter what the distribution. Similarly,
the optimal-pegging number of a graph is defined as the smallest distribution
of pegs for which all the vertices in the graph can be reached. We obtain tight
bounds on the pegging numbers and optimal-pegging numbers of complete binary
trees and compute the optimal-pegging numbers of complete infinitary trees. As
a result of these computations, we deduce that there is a tree whose
optimal-pegging number is strictly increased by removing a leaf. We also
compute the optimal-pegging number of caterpillar graphs and the tightest upper
bound on the optimal-pegging numbers of lobster graphs.
