 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 givesAt $R=5/4$, the upper-branch risk-neutral probabilities are $(RS-S_d)/(S_u-S_d)$. They are $3/8$ 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 $1/16,5/16,25/48,5/48$.