This is weird... the plant tag on the Hornbeam (from Miniature Plant Kingdom) says zone 6, but when I went to their site, they said C. heryana is hardy to zone 4. So maybe I won;t worry quite so much!

Ok, I think the discrepancy comes in when you look up how cold hardy hornbeam is in general, and see zone 4. However, if you put that tree in a tiny pot, it immediately loses a lot of its hardiness as it doesn't have the ground to protect it, nor other trees, etc. Mostly it's about the 1-2" of soil in a pot just not being nearly enough to keep it warm in winter, however it's also necessary for it to get cold and go dormant to survive any length of time. That's why you would, in ~ November, sink the pot into a much larger one full of mulch (with only the tree-above-soil uncovered, hopefully to be covered with snow, which is insulating. Got it? Great!

Lucy - Would sinking it into the garden (with snow cover, we hope) be preferable to sinking it into a pot and sticking it in the shed (no snow cover)? I could also throw a little salt marsh hay over it, in either spot.

It would be a better idea, but only if it's already been growing outdoors all year now, because otherwise it'll be shocked and it would be better to wait at least a couple of wks. It'll grow faster too in the ground, but only if you leave it 2-3 yrs - the first year is often about recovering from the move (plants are slow!) rather than producing much.

Just twigged to your other comment - I hope that hay doesn't actually have any salt in it - it would be deadly to your tree, though some species can stand tiny bits if living near the ocean naturally.

saltmarsh hay usually doesn;t have much in the way of salt in it - it's used as a winter mulch because whatever weed seeds come with it won't germinate/thrive in normal garden soil, unlike regular hay or straw. I have never heard of anyone suffering from salt damage through its use.