▁not ▁have ▁any ▁terminal ▁nodes ▁is ▁called ▁pr un ed . ▁ ▁Rel ation ▁to ▁other ▁types ▁of ▁trees ▁In ▁graph ▁theory , ▁a ▁root ed ▁tree ▁is ▁a ▁directed ▁graph ▁in ▁which ▁every ▁vertex ▁except ▁for ▁a ▁special ▁root ▁vertex ▁has ▁exactly ▁one ▁out going ▁edge , ▁and ▁in ▁which ▁the ▁path ▁formed ▁by ▁following ▁these ▁edges ▁from ▁any ▁vertex ▁eventually ▁leads ▁to ▁the ▁root ▁vertex . ▁If ▁ ▁is ▁a ▁tree ▁in ▁the ▁descript ive ▁set ▁theory ▁sense , ▁then ▁it ▁corresponds ▁to ▁a ▁graph ▁with ▁one ▁vertex ▁for ▁each ▁sequence ▁in ▁, ▁and ▁an ▁out going ▁edge ▁from ▁each ▁non empty ▁sequence ▁that ▁connect s ▁it ▁to ▁the ▁shorter ▁sequence ▁formed ▁by ▁removing ▁its ▁last ▁element . ▁This ▁graph ▁is ▁a ▁tree ▁in ▁the ▁graph - the or etic ▁sense . ▁The ▁root ▁of ▁the ▁tree ▁is ▁the ▁empty ▁sequence . ▁ ▁In ▁order ▁theory , ▁a ▁different ▁notion ▁of ▁a ▁tree ▁is ▁used : ▁an ▁order - the or etic ▁tree ▁is ▁a ▁partially ▁ordered ▁set ▁with ▁one ▁minimal ▁element ▁in ▁which ▁each ▁element ▁has ▁a ▁well - ordered ▁set ▁of ▁prede cess ors . ▁Every ▁tree ▁in ▁descript ive ▁set ▁theory ▁is ▁also ▁an ▁order - the or etic ▁tree , ▁using ▁a ▁partial ▁ordering ▁in ▁which ▁two ▁sequences ▁ ▁and ▁ ▁are ▁ordered ▁by ▁ ▁if ▁and ▁only ▁if ▁ ▁is ▁a ▁proper ▁prefix ▁of ▁. ▁The ▁empty ▁sequence ▁is ▁the ▁unique ▁minimal ▁element , ▁and ▁each ▁element ▁has ▁a ▁finite ▁and ▁well - ordered ▁set ▁of