To analyze the compound noun "Greek history teacher" and the provided tree diagrams, we need to understand the hierarchy and structure of noun phrases.

The compound noun can be interpreted in different ways based on how the words are grouped together.

1. Diagram I Analysis:
   • $N_1$ combines "Greek" and a further noun phrase $N_3$.
   • $N_3$ is a noun phrase combining "history" ($N_4$) and "teacher" ($N_5$).
   • In this interpretation, "teacher" is the head of both $N_3$ and the entire phrase $N_1$ which indicates a "teacher of Greek history."
2. Diagram II Analysis:
   • $N_1$ is formed by combining $N_2$ (which itself consists of "Greek" and "history") and "teacher" ($N_3$).
   • Here, "history" serves as the head of $N_2$, and "teacher" is the head of $N_1$.
   • This structure suggests a "teacher of history who is Greek."

Now, let's evaluate the given options to determine their correctness:

1. In diagram (I), ‘teacher’ is the head of both $N_1$ and $N_3$. — Correct. Explanation provided above confirms this.
2. In diagram (II), ‘history’ is the head of $N_2$, and ‘teacher’ is the head of $N_1$. — Correct. The analysis confirms this interpretation as well.
3. In diagram (I), ‘Greek’ is the head of $N_1$, and ‘teacher’ is the head of $N_3$. — Incorrect. "Teacher" is the head of $N_1$ in diagram (I).
4. In diagram (II), ‘teacher’ is the head of both $N_1$ and $N_2$. — Incorrect. In diagram (II), "teacher" is not the head of $N_2$ ("history" is).

Conclusion: Options 1 and 2 are the correct statements. Both diagrams provide different interpretations of the same compound noun based on the hierarchical structure.