Runner beans will climb strings.
I'd look for a large and established tree with strong limbs. Prune the limbs back a bit on the sunny side to let light in, then tie a string from a sturdy branch hanging down to the ground. If you use a biodegradable twine you can just cut top and bottom in the autumn and throw the whole lot on the
compost, or let it fall and rot in place.
I suspect that this will work better than trying to train up trunks, as on large strong trees the trunks are usually shaded and on younger trees the runner bean may swamp the tree.
I haven't tried this on trees, but I have made a sturdy wooden scaffold to hang strings from and they climb it with ease.