Not totally true, the Lumber Axe has a cutoff at 30 blocks without leaves. If your tree is less then 31 blocks tall and has leaf blocks on it like a normal tree, then enjoy your charcoal rain.
As a secondary note, Reika's Dye Trees has a type of tree called a Rainbow Tree, that is less then 31 blocks tall, and is made only of oak wood, but has a good sapling drop rate, okay apple drop rate, godly dye drop rate, occasionally drops golden apples, and rarely drops enchanted golden apples. One Rainbow tree averages me a full stack of every type of dye, two stacks of oak logs, 16 or so apples, 5 golden apples, and 1 or two enchanted golden apples.
My god... can it be farmed?