A stock index currently stands at $34$ and has a volatility of $30\%.$ The continuous compounding risk$-$free interest rate is $8\%$ and the continuous compounding dividend yield on the index is $3\%.$ Use a three$-$period binomial tree to value a six$-$month put option on the index with a strike price of $35$

$(4\mathrm{.}1)$ Assume the option is European.

$(4\mathrm{.}2)$ Assume the option is American.