Two variations of the intersection type assignment system are studied
in connection with B\"ohm-trees.  One is the intersection type
assignment system with a non-standard subtype relation, by means of
which we characterize whereabouts of $\bot$ in B\"ohm-trees.  The
other is a refinement of the intersection type assignment system whose
restricted typability is shown to coincide with finiteness of
B\"ohm-trees.