We settle a recent conjecture on a continuous patrolling game. In this
zero-sum game, an Attacker chooses a time and place to attack a network for a
fixed amount of time. A Patroller patrols the network with the aim of
intercepting the attack with maximum probability. The conjecture asserts that a
particular patrolling strategy called the E-patrolling strategy is optimal for
all tree networks. The conjecture was previously known to be true in a limited
class of special cases. The E-patrolling strategy has the advantage of being
straightforward to calculate and implement. We prove the conjecture by
presenting ε-optimal strategies for the Attacker which provide
upper bounds for the value of the game that come arbitrarily close to the lower
bound provided by the E-patrolling strategy