I'm trying to understand countable ordinals and their tree representation.
I understand that $\omega$ is the first "non branching tree" of infinite height.
I also understand that the exponent of $\omega$, so $\omega^2$ for example, corresponds to the node of the tree. So here we would have a tree of height $2$, for $\omega^3$ a tree of height $3$ etc.
So would the following two constructions be correct for $\omega \cdot 3$ (left) and $\omega^2$ (right):
