It is well-known that on a non-dividend paying stock, it is suboptimal to exercise an American call option earlier. In other words, both European and American call options on the same non-dividend paying stock worth the same. This can be proven using the Put-Call Parity.
However, I am not sure about European and American put options. I know that due to the ability that an American option can be exercised any time prior to maturity, it should worth at least as much as European put option. I am interested to know the conditions that give equality. In particular,
Question: Under what conditions will both European and American put options worth the same?
My motivation behind asking this question is so that I can ensure that my binomial tree implementation to price American put option is correct.
When dividend is zero, my binomial output the same price for both European and American call options, which is a good sign. However, I have no alternative to check whethee the tree gives correct price for American put option or not.