Please ignore my answer I'm not sure.

The holder of an American put option chooses to exercise this option at expiry. Which of the following must be true? This American put and an otherwise identical European put must have the same values at each node of a binomial model. At expiry the holder of the American option will receive the same payoff as the holder of an otherwise identical European option. The premium of this American put must be equal to the premium of an otherwise identical European put. The holder has made a mistake because American puts should be exercised prior to expiry