// Textbook fragment 10.10
  /** Creates a new tree node. */
  protected BTPosition<Entry<K,V>> createNode(Entry<K,V> element,
      BTPosition<Entry<K,V>> parent, BTPosition<Entry<K,V>> left,
      BTPosition<Entry<K,V>> right) {
    return new RBNode<K,V>(element,parent,left,right); // a red-black node
  }
  public Entry<K,V> insert(K k, V x) throws InvalidKeyException  {
    Entry<K,V> toReturn = super.insert(k, x);
    Position<Entry<K,V>> posZ = actionPos; // start at the insertion position
    setRed(posZ);
    if (isRoot(posZ))
      setBlack(posZ);
    else
      remedyDoubleRed(posZ); // fix a double-red color violation
    return toReturn;
  }
  protected void remedyDoubleRed(Position<Entry<K,V>> posZ)  {
    Position<Entry<K,V>> posV = parent(posZ);
    if (isRoot(posV))
      return;
    if (!isPosRed(posV))
      return;
    // we have a double red: posZ and posV
    if (!isPosRed(sibling(posV)))  { // Case 1: trinode restructuring
      posV = restructure(posZ);
      setBlack(posV);
      setRed(left(posV));
      setRed(right(posV));
    }
    else  { // Case 2: recoloring
      setBlack(posV);
      setBlack(sibling(posV));
      Position<Entry<K,V>> posU = parent(posV);
      if (isRoot(posU))
        return;
      setRed(posU);
      remedyDoubleRed(posU);
    }
  }