"If a tree falls in a forest and no one is around to hear it, does it make a sound?"
"No, I cast a permanent silence on it"
"Ah...right, that settles that then."
In seriousness, a continual item of silence was mentioned before... but surely you could make it a smaller area, i.e. 5ft/1 square so it only fills the tree's space and it would actually be cheaper? due to being a more limited version of the spell.
As long as it's within 5ft of the tree's "mouth" then it'd work... Hell if you made it into a pearl, like a pearl of speech, but a pearl of silence, force it into the lumberjack's mouth and then do the tree curse. so then it should only effect the mouth, and he can't take it out.
Cursed Pearl of Speech?