Tosefta do Horajot 3:14

These are the differences between the High Priest and a common priest: The bull that is offered for [the inadvertent transgression of] any commandment, and the bull brought on Yom Kippur, and the tenth of the ephah (Hor. 3:4). [The High Priest does not] grow his hair long [when in mourning] and [he] does not rend his garments [in the way that ordinary mourners do]. But [the High Priest] rends his garments from below [when in mourning], and the common priest [rends his garments] from above. [The High Priest] is not rendered impure by the death of his close relatives; and he is commanded to marry a virgin; and his is forbidden to marry a widow; and [his death permits] the return of the [accidental] killer [from a city of refuge], and he may sacrifice [when he is] an onen (a mourner on the first day of mourning, before the burial), but he may not eat from sacrifices or take a portion of them [on that day]; and he has the first [right] to sacrifice; and he takes his portion [of the sacrifice] first; and all the services performed on Yom Kippur are not valid unless [performed] by him; and he is exempt from [bringing an offering for inadvertently rendering] impure the Temple or the sacred utensils. And all of these (i.e., the aforementioned list) are conducted by a former [High Priests wearing] many vestments [and not only to High Priests who were anointed, who served during the First Temple period]; except as to the bull offered for [the inadvertent transgression of] any commandment, and none of them apply to an anointed [High Priest] for war, except for the five things that were explained, as written in the chapter [on the High Priest, Lev. 21:10-15]: he does not let his hair grow long [when in mourning]; he does not rend [his garments in the way that ordinary mourners do]; he is not rendered impure by the death of his close relatives; and he is commanded to marry a virgin; and his is forbidden to marry a widow; and [his death permits] the return of the [accidental] killer [from a city of refuge], the words of Rabbi Meir. And the Sages say, [his death does not permit] the return of the [accidental] killer.

Who is the nasi [that is obligated to bring an offering for sins committed before his appointment (Hor. 3:3, Hor. 10a:6)]? [He is] the "nasi" (per Vienna Man.) of Israel (i.e., the king), not the "nasi" of a tribe (i.e., a tribal leader, cf. Num. 7:2). [In the event that there is both] a "nasi of Israel" and a "nasi of the House of David" -- this one brings [a sacrifice of] his own, and that one brings [a sacrifice of] his own.