Where does “woodshed” come from?

I need to work through this chain carefully to create an accurate etymology sentence. The chain given is: English wood | English shade | Old English wudu | Proto-Indo-European weh₂t- | Proto-Indo-European skÁit- | Proto-Indo-European skÁit- | Proto-Indo-European weydʰ- | Proto-Indo-European sek- However, I notice this chain seems problematic. "Woodshed" is a compound of "wood" + "shed." The chain provided appears to conflate multiple separate etymologies. A "shed" comes from different roots than "wood." Given the constraint that I "state only what the chain supports" and "do not invent," and given that this chain doesn't clearly support a coherent path for "woodshed" as a compound word, I should note that the chain as presented doesn't provide a reliable basis for

woodshed (English): An enclosed, roofed structure, often an...