In a previous comment I saw that "cây" is for thin and long wooden objects like a stick, a cane or even a tree. But a bridge, what the heck
It doesn't matter.
U better learn it by heart ;) We use different classifier for one object either.
cây can also be a classifier for some musical instruments, particularly string instruments.
Monkey bridges are made out of big sticks to walk on.
Maybe "cây" is a wooden things classifier?
I'd rather say long, usually but not exclusively cylindrical, objects.