A variant of stack in which one other cactus stack may be attached to the top. An attached stack is called a branch. When a branch becomes empty, it is removed. Pop is not allowed if there is a branch. A branch is only accessible through the original reference. It is not accessible through the stack. It is also called a saguaro (a kind of branching cactus) stack or spaghetti stack.

Note: The operations new to this variant of stack, branch(v, u) and notch(v), may be defined with the following. top(branch(v, u)) = top(v)
not notch(new)
not notch(push(v, S))
notch(branch(v, u))