Mulberry trees are are a pretty easy tree to grow and chickens love the berries as well as the young leaves. Make sure you get a fruitinh variety because there's a lot of fruitless varities that they sell as landscape trees for the kind of people who find fruit to be a nuisance.
"Instead of Pay It Forward I prefer Plant It Forward" ~Howard Story / "God has cared for these trees, saved them from drought, disease, avalanches, and a thousand tempests and floods. But he cannot save them from fools." ~John Muir
I second the mulberry idea, particularly black mulberries. The nigra species is much sweeter and juicier. I've never used currants and gooseberries integrated into chicken forage, but both of them grow well and will produce a bit even in the shade so I'd also recommend trying those.