Suppose we have a power tower consisting of $2$ occurring $n$ times:

$$\huge2^{2^{2^{.^{.^{.^{2}}}}}}$$

How many values can we generate by placing any number of parenthesis?

It is fairly simple for the first few values of $n$:

- There is $1$ value for $n=1$:
- $2=2$

- There is $1$ value for $n=2$:
- $4=2^{2}$

- There is $1$ value for $n=3$:
- $16=({2^{2})^{2}}=2^{(2^{2})}$

- There are $2$ values for $n=4$:
- $256=(({2^{2})^{2}})^2=(2^{(2^{2})})^2=(2^{2})^{(2^{2})}$
- $65536=2^{(({2^{2})^{2}})}=2^{(2^{(2^{2})})}$

Any idea how to formulate a general solution?

I'm thinking that it might be feasible using a recurrence relation.

Thanks