// Textbook fragment 10.04
/** Checks whether a given key is valid. */
protected void checkKey(K key) throws InvalidKeyException {
  if(key == null)  // just a simple test for now
    throw new InvalidKeyException("null key");
}
/** Checks whether a given entry is valid. */
protected void checkEntry(Entry<K,V> ent) throws InvalidEntryException {
  if(ent == null || !(ent instanceof BSTEntry))
    throw new InvalidEntryException("invalid entry");
}
/** Auxiliary method for inserting an entry at an external node */
protected Entry<K,V> insertAtExternal(Position<Entry<K,V>> v, Entry<K,V> e) {
  expandExternal(v,null,null);
  replace(v, e);
  numEntries++;
  return e;
}
/** Auxiliary method for removing an external node and its parent */
protected void removeExternal(Position<Entry<K,V>> v) {
  removeAboveExternal(v);
  numEntries--;
}
/** Auxiliary method used by find, insert, and remove. */
protected Position<Entry<K,V>> treeSearch(K key, Position<Entry<K,V>> pos) {
  if (isExternal(pos)) return pos; // key not found; return external node
  else {
    K curKey = key(pos);
    int comp = C.compare(key, curKey);
    if (comp < 0)
      return treeSearch(key, left(pos));        // search left subtree
    else if (comp > 0)
      return treeSearch(key, right(pos));       // search right subtree
    return pos;         // return internal node where key is found
  }
}
// Adds to L all entries in the subtree rooted at v having keys equal to k
protected void addAll(PositionList<Entry<K,V>> L,
                      Position<Entry<K,V>> v, K k) {
  if (isExternal(v)) return;
  Position<Entry<K,V>> pos = treeSearch(k, v);
  if (!isExternal(pos))  {  // we found an entry with key equal to k
    addAll(L, left(pos), k);
    L.addLast(pos.element());   // add entries in inorder
    addAll(L, right(pos), k);
  } // this recursive algorithm is simple, but it's not the fastest
}