See when early exercise of an American option beats holding it.
At every node, compare the exercise value (intrinsic payoff today) with the continuation value (discounted risk-neutral expectation of waiting). The American holder takes the larger of the two: max(exercise, continuation). Nodes where exercising wins are the early-exercise region.
Start from the deep-ITM put and slowly raise the spot toward the strike. The green region shrinks and the premium falls — why does early exercise stop paying as the put moves out of the money? Now switch to a call: no node ever turns green and the premium sits at ≈ 0. What does Merton (1973) say about a call on a non-dividend stock, and how would a discrete dividend change that answer?
Cox–Ross–Rubinstein lattice with one period per step (Δt = 1): u = eσ√Δt, d = 1/u, p = (erΔt − d)/(u − d). At each node continuation = e−rΔt[p·Vup + (1−p)·Vdown], exercise = max(payoff, 0), American = max(exercise, continuation). The European value reprices the same tree with no early-exercise test, so VAm ≥ VEualways holds. No dividends modelled.