When do leaves have the biggest chance of dropping saplings: when I break the leaves or when I just break the wood and let the leaves die out?!
Nothing affects the chance for sapling drops. Saplings have a chance to drop when a leaf block either decays naturally from destroying the tree, or when you manually break the leaf block yourself. Either method produces the same results with the same chance of a sapling drop.
It doesn't matter at all.
Someone on the Minecraft Forum even looked at the (decompiled?) source code to confirm it.
And there's a "myth busting" episode on this topic by Xisuma: