The structural necessary and sufficient condition for "the transition-liveness means the place-liveness and vice-versa" of a subclass NII of general Petri nets is given as "the place and transition live Petri net, or PTL net, ÑII". Furthermore, "the one-token-condition Petri net, or OTC net, II" which means that every MSDL (minimal structural deadlock) is "transition and place live" under at least one initial token, i.e., II is "transition and place live" under the above initial marking. These subclasses NII, ÑII( NII), and II(ÑII) are almost the general Petri nets except at least one MSTR(minimal structural trap) and at least one pair of "a virtual MSTR or a virtual STR" and "a virtual MSDL" of an MBTR (minimal behavioral trap) in connection with making an MSDL transition-live.