If Tree of Perdition isn’t on the battlefield when the ability resolves, the exchange can’t happen and the ability will have no effect. Notably, activating the ability and giving Tree of Perdition -13/-13 in response won’t cause your opponent to lose the game.

When the ability resolves, Tree of Perdition’s toughness becomes the targeted opponent’s former life total and that player gains or loses an amount of life necessary so that their life total equals Tree of Perdition’s former toughness. Other effects that interact with life gain or life loss will interact with this effect accordingly.

Any toughness-modifying effects, counters, Auras, or Equipment will apply after its toughness is set to the player’s former life total. For example, say Tree of Perdition is equipped with Cultist’s Staff (which makes it 2/15) and the player’s life total is 7. After the exchange, Tree of Perdition would be a 2/9 creature (its toughness became 7, which was then modified by Cultist’s Staff) and the player’s life total would be 15.