// Textbook fragment 07.16
/**
  * An implementation of the BinaryTree interface by means of a linked structure
.
  */
public class LinkedBinaryTree<E> implements BinaryTree<E> {
  protected BTPosition<E> root; // reference to the root
  protected int size;           // number of nodes
  /**  Creates an empty binary tree. */
  public LinkedBinaryTree() {
    root = null;  // start with an empty tree
    size = 0;
  }
  /** Returns the number of nodes in the tree. */
  public int size() {
    return size;
  }
  /** Returns whether a node is internal. */
  public boolean isInternal(Position<E> v) throws InvalidPositionException {
    checkPosition(v);           // auxiliary method
    return (hasLeft(v) || hasRight(v));
  }
  /** Returns whether a node is the root. */
  public boolean isRoot(Position<E> v) throws InvalidPositionException {
    checkPosition(v);
    return (v == root());
  }
  /** Returns whether a node has a left child. */
  public boolean hasLeft(Position<E> v) throws InvalidPositionException {
    BTPosition<E> vv = checkPosition(v);
    return (vv.getLeft() != null);
  }
  /** Returns the root of the tree. */
  public Position<E> root() throws EmptyTreeException {
    if (root == null)
      throw new EmptyTreeException("The tree is empty");
    return root;
  }
  /** Returns the left child of a node. */
  public Position<E> left(Position<E> v)
    throws InvalidPositionException, BoundaryViolationException {
    BTPosition<E> vv = checkPosition(v);
    Position<E> leftPos = vv.getLeft();
    if (leftPos == null)
      throw new BoundaryViolationException("No left child");
    return leftPos;
  }