Are all binary operations on this binary tree distinct?

Question

Consider the set $\mathbb{Z}_+$ of positive integers $\{1,2,3,4,...\}$. Consider the binary operation $*$ of exponentiation on that set. I now define an infinite binary tree, constructed recursively as follows. The root of the tree is labelled with exponentiation. The left successor of the root is labeled with this binary operation $f : f(x,n) = (...(x*x)*x)...*x)$, where there are $n$ $x$'s. The right successor of the root is labeled with this binary operation $g: g(x,n) = (x*(x*...(x*x)...)$, where there are $n$ $x$'s. Basically, $g$ is tetration, and $f$ is the "left-associated" version of tetration. For example, $f(2,4)=(((2^2)^2)^2)=256$. Now, to continue the tree, we basically do the same thing we did with the left successor and right successor of the root. We label the left successor with the left-associated version of the operation iterated $n$ times, and we label the right successor with the right-associated version of the operation iterated $n$ times. This recursive procedure gives rise to a full infinite labelled binary tree, whose root is labelled with exponentiation. My question is, are there any "duplicate" labels in this tree? Meaning, are there distinct nodes of the full infinite binary tree which are in fact labelled with the exact same binary operation? I would be very surprised if there were. But I would like to see a rigorous proof that there aren't duplicate labels.