Why is Direct Form 2 (Digital Filters) equivalent to Direct Form 1
The word "order" is used two ways in your quotation from wikipaedia: -


  The alternate Direct Form II only needs N delay units, where N is the
  order of the filter


and


  This structure is obtained by reversing the order of the numerator and
  denominator sections of Direct Form I


In the first quote "order" refers to the "order" of the filter i.e. 1st order, 2nd order etc..

In the 2nd quote, "reversing the order" refers to a re-arrangement of the "circuit": -



Becomes this: -



These are both 2nd order filters and the re-arrangement (reversing of order) of the "a" coefficients with the "b" coefficients (in the circuit) is just a simplification. If you look at the two pictures hard enough you can see how this is achieved: -



Finally the two sections of delay elements (\$Z^{-1}\$) can be shared. No need for math!!