// Textbook fragment 08.11
  /** Returns whether v is an internal node. */
  public boolean isInternal(Position<E> v) throws InvalidPositionException {
    return hasLeft(v);  // if v has a right child it will have a left child
  }
  /** Returns whether v is an external node. */
  public boolean isExternal(Position<E> v) throws InvalidPositionException {
    return !isInternal(v);
  }
  /** Returns whether v is the root node. */
  public boolean isRoot(Position<E> v) throws InvalidPositionException {
    BTPos<E> vv = checkPosition(v);
    return vv.index() == 1;
  }
  /** Returns whether v has a left child. */
  public boolean hasLeft(Position<E> v) throws InvalidPositionException {
    BTPos<E> vv = checkPosition(v);
    return 2*vv.index() <= size();
  }
  /** Returns whether v has a right child. */
  public boolean hasRight(Position<E> v) throws InvalidPositionException {
    BTPos<E> vv = checkPosition(v);
    return 2*vv.index() + 1 <= size();
  }
  /** Returns the root of the tree. */
  public Position<E> root() throws EmptyTreeException {
    if (isEmpty()) throw new EmptyTreeException("Tree is empty");
    return T.get(1);
  }
  /** Returns the left child of v. */
  public Position<E> left(Position<E> v)
    throws InvalidPositionException, BoundaryViolationException {
    if (!hasLeft(v)) throw new BoundaryViolationException("No left child");
    BTPos<E> vv = checkPosition(v);
    return T.get(2*vv.index());
  }
  /** Returns the right child of v. */
  public Position<E> right(Position<E> v)
    throws InvalidPositionException {
    if (!hasRight(v)) throw new BoundaryViolationException("No right child");
    BTPos<E> vv = checkPosition(v);
    return T.get(2*vv.index() + 1);
  }