= Solution
{c}
Retain the whole price history when calculating the arithmetic average. The <Asian option> payoffs at expiry are
$$
\boxed{\begin{array}{c|c|c|c}
\text{path}&(S_0,S_1,S_2)&A_2&(115-A_2)^+\\\hline
uu&(100,120,144)&364/3&0\\
ud&(100,120,120)&340/3&5/3\\
du&(100,100,120)&320/3&25/3\\
dd&(100,100,100)&100&15
\end{array}}
$$
In particular $ud$ and $du$ have the same terminal stock price but different averages and payoffs. The average is an additional state variable; the terminal stock nodes cannot be merged for this claim.
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