Using only the definition of a binary search tree, show that if a node in a binary search tree has two children, then its successor has no left child and that its predecessor has no right child. Here all the keys in the binary search tree are distinct. The successor of a node j is defined as follows: if the node j has the biggest key, the successor is NIL; otherwise the successor is the node i that has a key that is bigger than the key of j and such that no other node has key between the key of i and the key of j.