// based on: https://algs4.cs.princeton.edu/33balanced/RedBlackBST.java.html
// TODO: implement NavigableSet
sclass UltraCompactTreeSet<A> extends AbstractSet<A> {

  // A symbol table implemented using a left-leaning red-black BST.
  // This is the 2-3 version.
 
  private static final boolean RED   = true;
  private static final boolean BLACK = false;

  private Node<A> root;     // root of the BST
  int size; // size of tree set

  // BST helper node data type
  abstract sclass Node<A> {
    A val;         // associated data
    Node<A> left, right;  // links to left and right subtrees

    abstract bool color();
    abstract BlackNode<A> convertToBlack();
    abstract RedNode<A> convertToRed();
    abstract Node<A> invertColor();
    Node<A> convertToColor(bool color) { ret color == RED ? convertToRed() : convertToBlack(); }
  }

  sclass BlackNode<A> extends Node<A> {
    *(A *val) {}
    *(A *val, Node<A> *left, Node<A> *right) {}
    
    bool color() { ret BLACK; }
    BlackNode<A> convertToBlack() { this; }
    RedNode<A> convertToRed() { ret new RedNode(val, left, right); }
    Node invertColor() { ret convertToRed(); }
  }

  sclass RedNode<A> extends Node<A> {
    *(A *val) {}
    *(A *val, Node<A> *left, Node<A> *right) {}

    bool color() { ret RED; }
    BlackNode<A> convertToBlack() { ret new BlackNode(val, left, right); }
    RedNode<A> convertToRed() { this; }
    Node invertColor() { ret convertToBlack(); }
  }
  
  *() {}
  *(Cl<? extends A> cl) { addAll(cl); }

  // is node x red; false if x is null ?
  static bool <A> isRed(Node<A> x) {
    ret x instanceof RedNode;
  }

  public int size() {
    ret size;
  }

  public bool isEmpty() {
    ret root == null;
  }

  public bool add(A val) {
    int oldSize = size;
    root = put(root, val);
    root = root.convertToBlack();
    ifdef CompactTreeSet_debug assertTrue(check()); endifdef
    ret size > oldSize;
  }

  // insert the value in the subtree rooted at h
  private Node<A> put(Node<A> h, A val) { 
    if (h == null) { ++size; ret new RedNode(val); }

    int cmp = compare(val, h.val);
    if      (cmp < 0) h.left  = put(h.left,  val); 
    else if (cmp > 0) h.right = put(h.right, val); 
    else              { /*h.val = val;*/ } // no overwriting

    // fix-up any right-leaning links
    if (isRed(h.right) && !isRed(h.left))      h = rotateLeft(h);
    if (isRed(h.left)  &&  isRed(h.left.left)) h = rotateRight(h);
    if (isRed(h.left)  &&  isRed(h.right))     h = flipColors(h);

    ret h;
  }
  
  // override me if you wish
  int compare(A a, A b) {
    ret cmp(a, b);
  }

  public bool remove(O key) { 
    if (!contains(key)) false;

    // if both children of root are black, set root to red
    if (!isRed(root.left) && !isRed(root.right))
        root = root.convertToRed();

    root = delete(root, (A) key);
    if (!isEmpty()) root = root.convertToBlack();
    // assert check();
    true;
  }

  // delete the key-value pair with the given key rooted at h
  private Node delete(Node<A> h, A key) { 
    // assert get(h, key) != null;

    if (compare(key, h.val) < 0)  {
      if (!isRed(h.left) && !isRed(h.left.left))
          h = moveRedLeft(h);
      h.left = delete(h.left, key);
    }
    else {
      if (isRed(h.left))
          h = rotateRight(h);
      if (compare(key, h.val) == 0 && (h.right == null)) {
          --size; null;
      } if (!isRed(h.right) && !isRed(h.right.left))
          h = moveRedRight(h);
      if (compare(key, h.val) == 0) {
          --size;
          Node<A> x = min(h.right);
          h.val = x.val;
          // h.val = get(h.right, min(h.right).val);
          // h.val = min(h.right).val;
          h.right = deleteMin(h.right);
      }
      else h.right = delete(h.right, key);
    }
    return balance(h);
  }

  // make a left-leaning link lean to the right
  private Node<A> rotateRight(Node<A> h) {
      // assert (h != null) && isRed(h.left);
      Node<A> x = h.left;
      h.left = x.right;
      x.right = h;
      x = x.convertToColor(x.right.color());
      x.right = x.right.convertToRed();
      ret x;
  }

  // make a right-leaning link lean to the left
  private Node<A> rotateLeft(Node<A> h) {
      // assert (h != null) && isRed(h.right);
      Node<A> x = h.right;
      h.right = x.left;
      x.left = h;
      x = x.convertToColor(x.left.color());
      x.left = x.left.convertToRed();
      ret x;
  }

  // flip the colors of a node and its two children
  private Node<A> flipColors(Node<A> h) {
      // h must have opposite color of its two children
      // assert (h != null) && (h.left != null) && (h.right != null);
      // assert (!isRed(h) &&  isRed(h.left) &&  isRed(h.right))
      //    || (isRed(h)  && !isRed(h.left) && !isRed(h.right));
      h.left = h.left.invertColor();
      h.right = h.right.invertColor();
      ret h.invertColor();
  }

  // Assuming that h is red and both h.left and h.left.left
  // are black, make h.left or one of its children red.
  private Node<A> moveRedLeft(Node<A> h) {
      // assert (h != null);
      // assert isRed(h) && !isRed(h.left) && !isRed(h.left.left);

      h = flipColors(h);
      if (isRed(h.right.left)) { 
          h.right = rotateRight(h.right);
          h = rotateLeft(h);
          h = flipColors(h);
      }
      ret h;
  }

  // Assuming that h is red and both h.right and h.right.left
  // are black, make h.right or one of its children red.
  private Node<A> moveRedRight(Node<A> h) {
      // assert (h != null);
      // assert isRed(h) && !isRed(h.right) && !isRed(h.right.left);
      h = flipColors(h);
      if (isRed(h.left.left)) { 
          h = rotateRight(h);
          h = flipColors(h);
      }
      ret h;
  }

  // restore red-black tree invariant
  private Node<A> balance(Node<A> h) {
      // assert (h != null);

      if (isRed(h.right))                      h = rotateLeft(h);
      if (isRed(h.left) && isRed(h.left.left)) h = rotateRight(h);
      if (isRed(h.left) && isRed(h.right))     h = flipColors(h);

      ret h;
  }


  /**
   * Returns the height of the BST (for debugging).
   * @return the height of the BST (a 1-node tree has height 0)
   */
  public int height() {
      ret height(root);
  }
  
  private int height(Node<A> x) {
      if (x == null) return -1;
      return 1 + Math.max(height(x.left), height(x.right));
  }
  
  public bool contains(O val) {
    ret find(root, (A) val) != null;
  }
  
  public A find(A probeVal) {
    Node<A> n = find(root, probeVal);
    ret n == null ? null : n.val;
  }

  // value associated with the given key in subtree rooted at x; null if no such key
  private A get(Node<A> x, A key) {
    x = find(x, key);
    ret x == null ? null : x.val;
  }

  Node<A> find(Node<A> x, A key) {
    while (x != null) {
      int cmp = compare(key, x.val);
      if      (cmp < 0) x = x.left;
      else if (cmp > 0) x = x.right;
      else              ret x;
    }
    null;
  }

  private boolean check() {
      if (!is23())             println("Not a 2-3 tree");
      if (!isBalanced())       println("Not balanced");
      return is23() && isBalanced();
  }

  // Does the tree have no red right links, and at most one (left)
  // red links in a row on any path?
  private boolean is23() { return is23(root); }
  private boolean is23(Node<A> x) {
      if (x == null) true;
      if (isRed(x.right)) false;
      if (x != root && isRed(x) && isRed(x.left)) false;
      ret is23(x.left) && is23(x.right);
  } 

  // do all paths from root to leaf have same number of black edges?
  private bool isBalanced() { 
      int black = 0;     // number of black links on path from root to min
      Node<A> x = root;
      while (x != null) {
          if (!isRed(x)) black++;
          x = x.left;
      }
      ret isBalanced(root, black);
  }

  // does every path from the root to a leaf have the given number of black links?
  private boolean isBalanced(Node<A> x, int black) {
      if (x == null) return black == 0;
      if (!isRed(x)) black--;
      return isBalanced(x.left, black) && isBalanced(x.right, black);
  }
  
  public void clear() { root = null; size = 0; }
  
  // the smallest key in subtree rooted at x; null if no such key
  private Node<A> min(Node<A> x) { 
    // assert x != null;
    while (x.left != null) x = x.left;
    ret x;
  }
  
  private Node<A> deleteMin(Node<A> h) { 
    if (h.left == null)
      return null;

    if (!isRed(h.left) && !isRed(h.left.left))
      h = moveRedLeft(h);

    h.left = deleteMin(h.left);
    ret balance(h);
  }
  
  public Iterator<A> iterator() {
    ret new MyIterator;
  }

  class MyIterator extends ItIt<A> {
    new L<Node<A>> path;
    
    *() {
      fetch(root);
    }
    
    void fetch(Node<A> node) {
      while (node != null) {
        path.add(node);
        node = node.left;
      }
    }
    
    public bool hasNext() { ret !path.isEmpty(); }

    public A next() {
      if (path.isEmpty()) fail("no more elements");
      Node<A> node = popLast(path);
      // last node is always a leaf, so left is null
      // so proceed to fetch right branch
      fetch(node.right);
      ret node.val;
    }
  }
  
  // Returns the smallest key in the symbol table greater than or equal to {@code key}.
  public A ceiling(A key) {
    Node<A> x = ceiling(root, key);
    ret x == null ? null : x.val;
  }

  // the smallest key in the subtree rooted at x greater than or equal to the given key
  Node<A> ceiling(Node<A> x, A key) {  
    if (x == null) null;
    int cmp = compare(key, x.val);
    if (cmp == 0) ret x;
    if (cmp > 0)  ret ceiling(x.right, key);
    Node<A> t = ceiling(x.left, key);
    if (t != null) ret t; 
    else           ret x;
  }
  
  public A floor(A key) {
    Node<A> x = floor(root, key);
    ret x == null ? null : x.val;
  }    

  // the largest key in the subtree rooted at x less than or equal to the given key
  Node<A> floor(Node<A> x, A key) {
    if (x == null) null;
    int cmp = compare(key, x.val);
    if (cmp == 0) ret x;
    if (cmp < 0)  ret floor(x.left, key);
    Node<A> t = floor(x.right, key);
    if (t != null) ret t; 
    else           ret x;
  }
}