// Textbook fragment 10.05
  // methods of the dictionary ADT
  public int size() { return numEntries; }
  public boolean isEmpty() { return size() == 0; }
  public Entry<K,V> find(K key) throws InvalidKeyException {
    checkKey(key);              // may throw an InvalidKeyException
    Position<Entry<K,V>> curPos = treeSearch(key, root());
    actionPos = curPos;         // node where the search ended
    if (isInternal(curPos)) return entry(curPos);
    return null;
  }
  public Iterable<Entry<K,V>> findAll(K key) throws InvalidKeyException {
    checkKey(key);              // may throw an InvalidKeyException
    PositionList<Entry<K,V>> L = new NodePositionList<Entry<K,V>>();
    addAll(L, root(), key);
    return L;
  }
  public Entry<K,V> insert(K k, V x) throws InvalidKeyException {
    checkKey(k);        // may throw an InvalidKeyException
    Position<Entry<K,V>> insPos = treeSearch(k, root());
    while (!isExternal(insPos))  // iterative search for insertion position
      insPos = treeSearch(k, left(insPos));
    actionPos = insPos; // node where the new entry is being inserted
    return insertAtExternal(insPos, new BSTEntry<K,V>(k, x, insPos));
  }
  public Entry<K,V> remove(Entry<K,V> ent) throws InvalidEntryException  {
    checkEntry(ent);            // may throw an InvalidEntryException
    Position<Entry<K,V>> remPos = ((BSTEntry<K,V>) ent).position();
    Entry<K,V> toReturn = entry(remPos);        // entry to be returned
    if (isExternal(left(remPos))) remPos = left(remPos);  // left easy case
    else if (isExternal(right(remPos))) remPos = right(remPos); // right easy ca
se
    else {                      //  entry is at a node with internal children
      Position<Entry<K,V>> swapPos = remPos;    // find node for moving entry
      remPos = right(swapPos);
      do
        remPos = left(remPos);
      while (isInternal(remPos));
      replaceEntry(swapPos, (Entry<K,V>) parent(remPos).element());
    }
    actionPos = sibling(remPos);        // sibling of the leaf to be removed
    removeExternal(remPos);
    return toReturn;
  }
}       // entries() method is omitted here