= Solution
At each nondegenerate binary node, no arbitrage requires the riskless gross return $R=1+r$ to lie strictly between the two stock returns. The three intervals are $(4/5,2)$ at time zero, $(6/5,3/2)$ at the upper node, and $(5/6,4/3)$ at the lower node. Their intersection gives
$$
\boxed{1/5<r<1/3.}
$$
At $R=5/4$, the upper-branch risk-neutral probabilities are $(RS-S_d)/(S_u-S_d)$. They are \b[$3/8$ initially, $1/6$ after the up move, and $5/6$ after the down move]. All are strictly between zero and one, so the <martingale> measure is equivalent and unique. The four terminal probabilities in the PDF's order $45,36,16,10$ are $1/16,5/16,25/48,5/48$.
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