I have lost count of how many branches I have carried for the goats over the years. Where I lived before, I only had a small paddock for the 6 goats I had, so every night I would go along the lanes and collect two big sacks full to keep them going the next day. They ate the branches, leaves, everything.
The goats I have now, only seem to want leaves, and leave the branches - the other goats loved to strip the bark off the willow, before eating the bare branch too.
Willow and ash are favourites, and they love the bay willow herb, the long plant with red flowers, that often grows on railway bankings. I am fortunate that we have plenty of tracks, and also a railway, with a continuous supply of trees and plants for the goats. On the railway bank is a lot of wild raspberry and red currant bushes ......so collect them too. I think a bird must have carried seeds from our garden up there, so they are really my bushes, and I am entitled to pick them!!