Let the dividend yield be $\delta$ and $C_u, C_d$ and $S_u, S_d$ be the up and down values for the stock and the call respectively over the period $\Delta t$.
In Hull and all other resources I've looked at, the hedge ratio stays the same in this case as the no dividend yield case, i.e. $$\Delta = \frac{C_u - C_d}{S_u - S_d}$$ which confuses me because the payoff of owning one share of stock is actually $S_u e^{\delta \Delta t}$ or $S_d e^{\delta \Delta t}$ so I would presume that the hedge ratio should shift to be $$\frac{C_u - C_d}{S_u - S_d} \exp (-\delta \Delta t)$$ Why is this not the case?
My attempt at a "reasonable" explanation:
Over short $\Delta t$ we have $\exp(\delta \Delta t) \approx (1+ \delta \Delta t)$ so that the approximate payoffs from the stock are $S_u + S \delta \Delta t$ in the up position and $S_d + S \delta \Delta t$ in the down position, where $S$ is the initial price of the stock, asymptotically, so we may just take the denominator in the hedge ratio to be $$S_u + S \delta \Delta t - (S_d + S \delta \Delta t) = S_u - S_d$$ as usual.