For example, I have a Howlgeist that I enchanted with Oath of Ancient Wood so it has a +1/+1 counter on it, when it dies does it not come back because of the +1/+1 counter even though it wasn't put there by Undying?
Undying doesn't care how the +1/+1 counter got there. Undying won't trigger.
Conversely, if Undying previously placed a +1/+1 counter on a permanent, but the counter got removed somehow before the permanent died a second time, Undying will trigger and bring the card back a second time.
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2It's important to note that putting a -1/-1 on to remove the +1/+1 can't be what actually causes leathal damage or 0 toughness. If it is, the state based action check will kill it with both counters still on it, so neither undying nor persist will trigger. – Michael Snook Nov 17 '14 at 2:48
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