Good morning, Dejan.

Here I have a summary of the argument concerning the Binary Tree:

1) The set of all real numbers of the unit interval is (said to be) uncountable.
2) An uncountable set has (infinitely many) more elements than a countable set.
3) Every real number has at least one unique representation as an infinite binary string (some rationals have even two representations but that's peanuts).
4) In many cases the string cannot be defined by a finite word.
5) Without loss of information the first bits of two strings, if equal, need not be written twice.