Parenthifiable Binary Numbers

Pyth, 10 bytes

!uscG`T.BQ

Try this test suite in the Pyth Compiler.

How it works

              (implicit) Store the evaluated input in Q.
       .BQ    Return the binary string representation of Q.
 u            Reduce w/base case; set G to .BQ and begin a loop:
     `T         Return str(10) = "10".
   cG           Split G (looping variable) at occurrences of "10".
  s             Join the pieces without separators.
              Set G to the returned string.
              If the value of G changed, repeat the loop.
              This will eventually result in either an empty string or a
              non-empty string without occurrences of "10".
!             Return (and print) the logical NOT of the resulting string.

TeaScript, 9 bytes 16 18 20 22 24

Saved 2 bytes thanks to @ETHproductions

!x÷W(n,¢)

Woah. That's short. Uses @xnor's approach. This will use the recursive replace function (W) which will replace all 10 equal to () with nothing. If the string is blank it is balanced.

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Using a version of TeaScript made after this challenge was posted this could become 7 bytes:

!x÷W(n)

Ungolfed

!xT(2)W(n,``)

Explanation

!      // NOT, returns true if empty string, else false
 xT(2)   // To binary
 W(n,``) // n is 10, reclusive replaces 10 or (), with nothing.

Python2, 87 bytes

try:exec"print 1,"+"".join(["],","["][int(c)]for c in bin(input())[2:])
except:print 0

A terrible implementation that abuses syntax errors.