A countable set can be constructed by using always half of the remaining time for the next step. An uncountable set cannot be constructed such that uncountably many elements can be distinguished. So it is possible to construct |N and with it all its subsets. But these subsets cannot be distinguished unless it is indicated which elements are to combine. Therefore we find:
- The Binary Tree can be constructed because it consists of countably many nodes and edges.
- The Binary Tree cannot be constructed because it consists of uncountably many distinct paths.