The Tree of Redemption is now a 0/1 blue frog with no abilities... but it's still the same object. For instance, if someone chucked a Doom Blade at it, and you cast Momentary Blink in response, the Tree leaves (pun intended) play and returns to play. Sure, it's the same card, probably on the same spot on the table, but for the game, it's a new object, so Doom Blade checks for that other Tree of the Redemption it was supposed to kill, fails to find it, and heads gloomly to the graveyard. If you cast Turn to Frog, on the other hand, Doom Blade sees the Tree trying to paint itself blue and don a frog costume, and still nails it. This explains MarcinS' answer, as once the ability is on the stack, it'll try to resolve and do as much as it can, so it'll start to resolve, find the Tree (it's still the same object, remember?) and do the life swap, only now it's toughness is 1.