For any Cartesian product (at least for finite number of sets), I can associate it with a tree where the leaves are the sets and each node is the Cartesian product of node directly below it. For example, $(A \times B \times C) \times (D \times (E \times F))$ is the root of the following tree
In fact, if $\prod_{i_1} \prod_{i_2} \cdots \prod_{i_n} X_{i_1 i_2 ... i_n}$ a finitely nested Cartesian product, then I can always associate such expression with a tree with finite height (if one accepts that a node can have infinite number of child nodes) and vice versa.
My question: is it possible to extend it to trees with infinite height? What would an infinitely nested Cartesian product look like?