. ▁ ▁properly ▁extends ▁), ▁or ▁ ▁both ▁ ▁and ▁ ▁are ▁defined , ▁, ▁and ▁. ▁Here , ▁the ▁notation ▁ ▁refers ▁to ▁the ▁prefix ▁of ▁ ▁up ▁to ▁but ▁not ▁including ▁. ▁In ▁simple ▁terms , ▁ ▁whenever ▁ ▁is ▁a ▁prefix ▁of ▁ ▁( i . e . ▁ ▁termin ates ▁before ▁, ▁and ▁they ▁are ▁equal ▁up ▁to ▁that ▁point ) ▁or ▁ ▁is ▁to ▁the ▁" left " ▁of ▁ ▁on ▁the ▁first ▁place ▁they ▁differ . ▁ ▁Tree ▁interpretation ▁A ▁tree , ▁in ▁descript ive ▁set ▁theory , ▁is ▁defined ▁as ▁a ▁set ▁of ▁finite ▁sequences ▁that ▁is ▁closed ▁under ▁prefix ▁operations . ▁The ▁parent ▁in ▁the ▁tree ▁of ▁any ▁sequence ▁is ▁the ▁shorter ▁sequence ▁formed ▁by ▁removing ▁its ▁final ▁element . ▁Thus , ▁any ▁set ▁of ▁finite ▁sequences ▁can ▁be ▁augment ed ▁to ▁form ▁a ▁tree , ▁and ▁the ▁Kle ene – Br ou wer ▁order ▁is ▁a ▁natural ▁ordering ▁that ▁may ▁be ▁given ▁to ▁this ▁tree . ▁It ▁is ▁a ▁general ization ▁to ▁potentially - in finite ▁trees ▁of ▁the ▁post order ▁travers al ▁of ▁a ▁finite ▁tree : ▁at ▁every ▁node ▁of ▁the ▁tree , ▁the ▁child ▁sub trees ▁are ▁given ▁their ▁left ▁to ▁right ▁ordering , ▁and ▁the ▁node ▁itself ▁comes ▁after ▁all ▁its ▁children . ▁The ▁fact ▁that ▁the ▁Kle ene – Br ou wer ▁order ▁is ▁a ▁linear ▁ordering ▁( that ▁is , ▁that ▁it ▁is ▁trans itive ▁as ▁well ▁as ▁being ▁total ) ▁follows ▁immediately ▁from ▁this , ▁as ▁any