// Textbook fragment 07.20
  /** Attaches two trees to be subtrees of an external node. */
  public void attach(Position<E> v, BinaryTree<E> T1, BinaryTree<E> T2)
    throws InvalidPositionException {
    BTPosition<E> vv = checkPosition(v);
    if (isInternal(v))
      throw new InvalidPositionException("Cannot attach from internal node");
    if (!T1.isEmpty()) {
      BTPosition<E> r1 = checkPosition(T1.root());
      vv.setLeft(r1);
      r1.setParent(vv);         // T1 should be invalidated
    }
    if (!T2.isEmpty()) {
      BTPosition<E> r2 = checkPosition(T2.root());
      vv.setRight(r2);
      r2.setParent(vv);         // T2 should be invalidated
    }
  }
  /** If v is a good binary tree node, cast to BTPosition, else throw exception
*/
  protected BTPosition<E> checkPosition(Position<E> v)
    throws InvalidPositionException {
    if (v == null || !(v instanceof BTPosition))
      throw new InvalidPositionException("The position is invalid");
    return (BTPosition<E>) v;
  }
  /** Creates a new binary tree node */
  protected BTPosition<E> createNode(E element, BTPosition<E> parent,
                                  BTPosition<E> left, BTPosition<E> right) {
    return new BTNode<E>(element,parent,left,right); }
  /** Creates a list storing the the nodes in the subtree of a node,
    * ordered according to the preorder traversal of the subtree. */
  protected void preorderPositions(Position<E> v, PositionList<Position<E>> pos)

    throws InvalidPositionException {
    pos.addLast(v);
    if (hasLeft(v))
      preorderPositions(left(v), pos);  // recurse on left child
    if (hasRight(v))
      preorderPositions(right(v), pos); // recurse on right child
  }