Consider the standard binary tree. Clearly the number of nodes is a countable infinity. Each node can be mapped bijectively to a rational number. But if we go ahead and "union the tree" with all "limiting nodes" which are all infinite binary sequences of $0$'s and $1$'s. In this sense we have now made the "collection of nodes" in the "completed tree" an uncountable infinity.
Is this "completed tree" still a tree, but just with a countable collection of finite nodes and an uncountable collection of "nodes at infinity"? If so, what is the proper mathematical way to describe it?
So, in an attempt to rephrase, the question is:
Can we "complete" the standard binary tree by adding an uncountable collection of nodes at "the end" and if so, then how do we describe it mathematically?
I envision the standard binary tree as being the countable collection of nodes. But can I "complete the tree" in this way by considering there to be a "level at $\omega$" with an uncountable collection of nodes? I envision this last level of nodes to still be smoothly connected to the branches converging towards it from the rest of the tree.
I can't quite make this rigorous because I think I lack the necessary set theory background, so I hope those with more expertise can see what I am going after here and to correct any errors or misconceptions I have. Maybe set theory is not the way to approach this, so any answer from any field is appreciated.