$12\mathrm{.}2-4$

Professor Kilmer claims to have discovered a remarkable property of binary search

trees. Suppose that the search for key $k$ in a binary search tree ends up at a leaf.

Consider three sets: $A,$ the keys to the left of the search path; $B,$ the keys on

the search path; and $C,$ the keys to the right of the search path. Professor Kilmer

claims that any three keys ainA,binB, and cinC must satisfy $abc.$ Give

a smallest possible counterexample to the professor's claim.