▁a ▁collection ▁of ▁finite ▁sequences ▁of ▁elements ▁of ▁ ▁such ▁that ▁every ▁prefix ▁of ▁a ▁sequence ▁in ▁the ▁collection ▁also ▁belongs ▁to ▁the ▁collection . ▁ ▁De fin itions ▁ ▁Tre es ▁The ▁collection ▁of ▁all ▁finite ▁sequences ▁of ▁elements ▁of ▁a ▁set ▁ ▁is ▁denoted ▁. ▁With ▁this ▁notation , ▁a ▁tree ▁is ▁a ▁non empty ▁subset ▁ ▁of ▁, ▁such ▁that ▁if ▁ ▁is ▁a ▁sequence ▁of ▁length ▁ ▁in ▁, ▁and ▁if ▁, ▁then ▁the ▁short ened ▁sequence ▁ ▁also ▁belongs ▁to ▁. ▁In ▁particular , ▁choosing ▁ ▁shows ▁that ▁the ▁empty ▁sequence ▁belongs ▁to ▁every ▁tree . ▁ ▁Branch es ▁and ▁bodies ▁A ▁branch ▁through ▁a ▁tree ▁ ▁is ▁an ▁infinite ▁sequence ▁of ▁elements ▁of ▁, ▁each ▁of ▁whose ▁finite ▁prefix es ▁belongs ▁to ▁. ▁The ▁set ▁of ▁all ▁branches ▁through ▁ ▁is ▁denoted ▁ ▁and ▁called ▁the ▁body ▁of ▁the ▁tree ▁. ▁ ▁A ▁tree ▁that ▁has ▁no ▁branches ▁is ▁called ▁well f ounded ; ▁a ▁tree ▁with ▁at ▁least ▁one ▁branch ▁is ▁ill f ounded . ▁ ▁By ▁K ő nig ' s ▁lemma , ▁a ▁tree ▁on ▁a ▁finite ▁set ▁with ▁an ▁infinite ▁number ▁of ▁sequences ▁must ▁necessarily ▁be ▁ill f ounded . ▁ ▁Terminal ▁nodes ▁A ▁finite ▁sequence ▁that ▁belongs ▁to ▁a ▁tree ▁ ▁is ▁called ▁a ▁terminal ▁node ▁if ▁it ▁is ▁not ▁a ▁prefix ▁of ▁a ▁longer ▁sequence ▁in ▁. ▁Equ ival ently , ▁ ▁is ▁terminal ▁if ▁there ▁is ▁no ▁element ▁ ▁of ▁ ▁such ▁that ▁that ▁. ▁ ▁A ▁tree ▁that ▁does