▁three ▁sequences ▁on ▁which ▁trans itivity ▁is ▁to ▁be ▁tested ▁form ▁( with ▁their ▁prefix es ) ▁a ▁finite ▁tree ▁on ▁which ▁the ▁Kle ene – Br ou wer ▁order ▁coinc ides ▁with ▁the ▁post order . ▁ ▁The ▁significance ▁of ▁the ▁Kle ene – Br ou wer ▁ordering ▁comes ▁from ▁the ▁fact ▁that ▁if ▁ ▁is ▁well - ordered , ▁then ▁a ▁tree ▁over ▁▁ ▁is ▁well - f ounded ▁( h aving ▁no ▁infinitely ▁long ▁branches ) ▁if ▁and ▁only ▁if ▁the ▁Kle ene – Br ou wer ▁ordering ▁is ▁a ▁well - order ing ▁of ▁the ▁elements ▁of ▁the ▁tree . ▁ ▁Rec urs ion ▁theory ▁In ▁recursion ▁theory , ▁the ▁Kle ene – Br ou wer ▁order ▁may ▁be ▁applied ▁to ▁the ▁computation ▁trees ▁of ▁implementations ▁of ▁total ▁recursive ▁function als . ▁A ▁computation ▁tree ▁is ▁well - f ounded ▁if ▁and ▁only ▁if ▁the ▁computation ▁performed ▁by ▁it ▁is ▁total ▁recursive . ▁Each ▁state ▁ ▁in ▁a ▁computation ▁tree ▁may ▁be ▁assigned ▁an ▁ord inal ▁number ▁, ▁the ▁supre m um ▁of ▁the ▁ord inal ▁numbers ▁ ▁where ▁ ▁ranges ▁over ▁the ▁children ▁of ▁ ▁in ▁the ▁tree . ▁In ▁this ▁way , ▁the ▁total ▁recursive ▁function als ▁themselves ▁can ▁be ▁class ified ▁into ▁a ▁hierarchy , ▁according ▁to ▁the ▁minimum ▁value ▁of ▁the ▁ord inal ▁at ▁the ▁root ▁of ▁a ▁computation ▁tree , ▁minim ized ▁over ▁all ▁computation ▁trees ▁that ▁implement ▁the ▁functional . ▁The ▁Kle ene – Br ou wer ▁order ▁of ▁a ▁well - f