European option and American option are equivalent in this case?

This is Question No.11 from 2007 May MFE Exam.

For a two-period binomial model for stock prices, you are given:

(1) Each period is 6 months.

(2) The current price for a nondividend paying stock is $70.00$

(3) $u=1.181$, $d=0.890$

(4) The continuously compounded risk-free interest rate is $5\%$.

Calculate the current price of a one-year American put option on the stock with strike price of $80.00$.

I supposed that for a nondividend paying stock, the price of American put option should be the same as the price of the corresponding European option. Following that thought, I constructed the binomial tree and my calculation is that

$24.553\times e^{-0.05} \times (1-0.465)^2+6.4237\times e^{-0.05}\times 2 \times (1-0.465)\times 0.465$

But I was reading the answer, and apparently when calculating the payoff at node $P_d$, the answer suggests it is optimal to early exercise the option.

$P_d=\max (K-S_d,e^{-rh} [P_{ud}p+P_{dd}(1-p)])=\max(80-62.30, e^{-0.05*0.5}[6.42\times 0.465+24.55\times(1-0.465)])=\max(17.70,15.72)$.

Is there anything wrong with the question? Or did I miss something? Thank you very much!

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