// Textbook fragment 08.12
  /** Returns the parent of v. */
  public Position<E> parent(Position<E> v)
    throws InvalidPositionException, BoundaryViolationException {
    if (isRoot(v)) throw new BoundaryViolationException("No parent");
    BTPos<E> vv = checkPosition(v);
    return T.get(vv.index()/2);
  }
  /** Replaces the element at v. */
  public E replace(Position<E> v, E o) throws InvalidPositionException {
    BTPos<E> vv = checkPosition(v);
    return vv.setElement(o);
  }
  /** Adds an element just after the last node (in a level numbering). */
  public Position<E> add(E e) {
    int i = size() + 1;
    BTPos<E> p = new BTPos<E>(e,i);
    T.add(i, p);
    return p;
  }
  /** Removes and returns the element at the last node. */
  public E remove() throws EmptyTreeException {
    if(isEmpty()) throw new EmptyTreeException("Tree is empty");
    return T.remove(size()).element();
  }
  /** Determines whether the given position is a valid node. */
  protected BTPos<E> checkPosition(Position<E> v)
    throws InvalidPositionException
  {
    if (v == null || !(v instanceof BTPos))
      throw new InvalidPositionException("Position is invalid");
    return (BTPos<E>) v;
  }
  /** Returns an iterator of the elements stored at all nodes in the tree. */
  public Iterator<E> iterator() {
    ArrayList<E> list = new ArrayList<E>();
    Iterator<BTPos<E>> iter = T.iterator();
    iter.next(); // skip the first element
    while (iter.hasNext())
      list.add(iter.next().element());
    return list.iterator();
  }
}