Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 4J / a / Solution

Take the productionsand the word $w=aa$. There is only one parse tree: its root uses $S\to AA$ and each child uses $A\to a$. There are nevertheless exactly two derivations,andaccording as the left or right occurrence of $A$ is replaced first. This illustrates that derivations record replacement order while a context-free grammar parse tree does not.