二叉查找树(BST)的实现

一、二叉树介绍

　　二叉查找树（Binary Search Tree，BST），又称二叉排序树，也称二叉搜索树，它或者是一颗空树，或者具有如下性质的树：若它的左子树不为空，则左子树上所有节点的值都小于根节点的值；若它的右子树不为空，则右子树上所有节点的值都大于根节点的值。它的左右子树也分别为二叉查找树。

　　结论：中序遍历一颗二叉查找树可以得到一个按关键字递增的有序序列。简单来说，比根小的往左边放  比根大的往右边放。

　　　　

二、代码实现

　　1、BST结点类：

public class BSTNode<K, V> {
    public K key;
    public V value;
    public BSTNode<K, V> left;
    public BSTNode<K, V> right;
    public BSTNode<K, V> parent;
    public boolean isLeftChild;
    public int height;
    public int num;
    public boolean isRed = true;  // 后面才学习的红黑树

    public BSTNode() {
    }

    public BSTNode(K key, V value, BSTNode<K, V> left, BSTNode<K, V> right, BSTNode<K, V> parent) {
        super();
        this.key = key;
        this.value = value;
        this.left = left;
        this.right = right;
        this.parent = parent;
    }

    public boolean isLeft() {
        return isLeftChild;
    }

    public boolean isRight() {
        return !isLeftChild;
    }

    @Override
    public String toString() {
        return (isRed ? "红色" : "黑色") + " [" + key + "]<-" + (parent == null ? "" : parent.key);
    }
}

　　2、BST接口：

import java.util.List;
import java.util.function.Consumer;

public interface IBinarySearchTree<K, V> {
    /**
     * 新增节点
     * @param k 关键字
     * @param v 值
     */
    BSTNode<K, V> insert(K k, V v);

    /**
     * 中序遍历
     * @param con 处理中序遍历的每个元素的函数
     */
    void inorder(Consumer<K> con);

    /**
     * 查找元素
     * @param key
     * @return
     */
    V lookupValue(K key);

    /**
     * 获取最小关键字
     * @return
     */
    K min();

    /**
     * 获取最大关键字
     * @return
     */
    K max();

    /**
     * 移除关键字对应的节点
     * @param key
     */
    void remove(K key);

    /**
     * x的后继——比x大的第一个元素 1、是其右子树的最小值
     * 2、没有右子树，则向上追溯，直到某个祖先节点是左孩子，返回这个祖先节点的父节点，它就是x的后继
     * 
     * @param x
     * @return
     */
    K successor(K x);

    /**
     * 前驱
     * @param x 关键字
     * @return
     */
    K predecessor(K x);

    boolean isBalance();

    /**
     * 返回节点数
     * @return
     */
    int getSize();

    /**
     * 高度
     * @return
     */
    int getHeight();

    List<List<BSTNode<K, V>>> levelOrder();
}

　  3、BST实现：

import java.util.ArrayList;
import java.util.Comparator;
import java.util.LinkedList;
import java.util.List;
import java.util.Queue;
import java.util.Stack;
import java.util.function.Consumer;

/**
 * 二叉搜索树
 *
 */
public class BinarySearchTree<K, V> implements IBinarySearchTree<K, V> {
  /**
   * 根节点
   */
  protected BSTNode root;
  /**
   * 元素个数
   */
  protected int size;
  private Comparator comparator;

  public BinarySearchTree() {
  }

  public BinarySearchTree(Comparator comparator) {
    this.comparator = comparator;
  }


  // parent curr 双指针
  @Override
  public BSTNode<K, V> insert(K key, V value) {
    if (!(key instanceof Comparable)) {
      throw new ClassCastException();
    }

    BSTNode<K, V> parent = null;
    BSTNode<K, V> curr = root;
    while (curr != null) {
      parent = curr;
      if (compare(key, curr.key) < 0) {
        curr = curr.left;
      } else if (compare(key, curr.key) > 0) {
        curr = curr.right;
      } else {
        curr.value = value;
        return curr;
      }
    }
    curr = new BSTNode(key, value, null, null, null);
    //link current to parent
    curr.parent = parent;
    if (parent == null) {
      root = curr;
    } else if (compare(key, parent.key) < 0) {
      parent.left = curr;
      curr.isLeftChild = true;
    } else {
      parent.right = curr;
      curr.isLeftChild = false;
    }

    size++;
    updateHeight(curr);
    return curr;
  }

  private void updateHeight(BSTNode<K, V> curr) {
    if (curr.parent == null) return;//util root

    BSTNode<K, V> p = curr.parent;
    if (p.height == curr.height) {
      p.height++;
      updateHeight(p);//递归
    }
  }

  @SuppressWarnings({"unchecked", "rawtypes"})
  private int compare(K key1, K key2) {
    if (null == comparator) {
      return ((Comparable) key1).compareTo((Comparable) key2);
    } else {
      return comparator.compare(key1, key2);
    }
  }

  /**
   * 中序遍历
   * @param con 处理中序遍历的每个元素的函数
   */
  @Override
  public void inorder(Consumer<K> con) {
    if (root != null)
      // inorder2(root, con);
      inorder(root, con);
  }

  /* 递归形式 */
  private void inorder(BSTNode<K, V> p, Consumer<K> con) {
    if (p != null) {
      inorder(p.left, con);
      con.accept(p.key);
      inorder(p.right, con);
    }
  }

  /*迭代形式*/
  private void inorder2(BSTNode<K, V> p, Consumer<K> con) {
    Stack<BSTNode<K, V>> stack = new Stack<>();
    BSTNode<K, V> curr = p;
    //curr不为空或者栈不为空，都可以继续处理
    while (curr != null || !stack.isEmpty()) {//没有生产也没有消费，就退出循环了
      //沿左支线一撸到底，全部入栈
      while (curr != null) {
        stack.push(curr);
        curr = curr.left;
      }
      //处理栈顶
      if (!stack.isEmpty()) {
        BSTNode<K, V> pop = stack.pop();
        con.accept(pop.key);
        //  curr指向pop的右子树，继续外层循环
        curr = pop.right;//有可能为空，为空，只消费栈中内容，不为空，就要向栈中生产若干内容
      }
    }
  }
  
  // 二叉查找树查找之所以快 每次丢弃一半  lgn
  @Override
  public V lookupValue(K key) {
    BSTNode<K, V> lookupNode = lookupNode(key);
    return lookupNode == null ? null : lookupNode.value;
  }

  protected BSTNode<K, V> lookupNode(K key) {
    BSTNode<K, V> p = root;
    //只要p不为空，并且没找到
    while (p != null && compare(key, p.key) != 0) {
      if (compare(key, p.key) < 0)
        p = p.left;
      else
        p = p.right;
    }
    return p;
  }

  // 最左边的结点
  @Override
  public K min() {
    return minNode(root).key;
  }

  protected BSTNode<K, V> minNode(BSTNode p) {
    while (p.left != null) {
      p = p.left;
    }
    return p;
  }

  // 最右边的结点
  @Override
  public K max() {
    return maxNode(root).key;
  }

  protected BSTNode<K, V> maxNode(BSTNode p) {
    while (p.right != null) {
      p = p.right;
    }
    return p;
  }

  /*右单旋
  *   p
  *   q
  *   */
  protected void rightRotate(BSTNode p, BSTNode q) {
    boolean pIsLeft = p.isLeft();
    BSTNode pp = p.parent;

    BSTNode x = q.right;
    p.left = x;
    if (x != null) {
      x.parent = p;
      x.isLeftChild = true;
    }
    q.right = p;
    p.parent = q;
    p.isLeftChild = false;


    //设定p和gg的关系
    q.parent = pp;
    if (pp == null) {
      root = q;
      return;
    }
    if (pIsLeft) {
      pp.left = q;
      q.isLeftChild = true;
    } else {
      pp.right = q;
      q.isLeftChild = false;
    }
  }

  /*左单旋*/
  protected void leftRotate(BSTNode p, BSTNode q) {
    boolean pIsLeft = p.isLeft();
    BSTNode pp = p.parent;
    //p和q的左子——B的关系
    BSTNode B = q.left;
    p.right = B;
    if (B != null) {
      B.parent = p;
      B.isLeftChild = false;
    }

    //p，q的关系
    q.left = p;
    p.parent = q;
    p.isLeftChild = true;

    //p和pp的关系
    q.parent = pp;
    //p是根节点
    if (pp == null) {
      root = q;
      return;
    }

    if (pIsLeft) {
      pp.left = q;
      q.isLeftChild = true;
    } else {
      pp.right = q;
      q.isLeftChild = false;
    }
  }

  @Override
  public void remove(K key) {
    removeNode(lookupNode(key));
    size--;// 记得减少元素个数
  }

  protected void removeNode(BSTNode<K, V> x) {
    if (x != null) {
      if (x.left == null && x.right == null) {// leaf node.  第一种情况 没有子节点
        if (x.parent == null) {
          root = null;
          return;
        }
        if (x.isLeftChild) {
          x.parent.left = null;
        } else {
          x.parent.right = null;
        }
        x.parent = null;
        x = null;
      } else if (x.left == null) {// 第二种情况 有子节点,但左子为空,有右孩子
        if (x.isLeftChild) {
          BSTNode<K, V> c = x.right;
          BSTNode<K, V> parent = x.parent;
          parent.left = c;
          c.isLeftChild = true;
          c.parent = parent;
        } else {
          if (x.parent != null) {
            x.parent.right = x.right;
            x.right.parent = x.parent;
          } else {// 根节点
            root = x.right;
          }
        }
        x = null;
      } else if (x.right == null) {// 第三种情况 有子节点,但右子为空，有左孩子
        if (x.isLeftChild) {
          x.parent.left = x.left;
          x.left.parent = x.parent;
        } else {
          if (x.parent != null) {
            x.parent.right = x.left;
            x.left.isLeftChild = false;
            x.left.parent = x.parent;
          } else { // 根节点
            root = x.left;
          }
        }
        x = null;
      } else {                // 第四种情况 都不为空
        BSTNode<K, V> minOfRight = minNode(x.right);
        x.key = minOfRight.key;// 更换x的内容
        removeNode(minOfRight); // 删掉右子树种最小的元素
      }
    }
  }


  // 有右子树 ，则后继为右子树最小
  // 否则往上回溯，找到一个是左结点的祖先，则后继是该结点的父亲
  @Override
  public K successor(K x) {
    BSTNode<K, V> xNode = lookupNode(x);
    if (xNode == null) {
      return null;
    }
    BSTNode<K, V> yNode = successor(xNode);
    return yNode == null ? null : yNode.key;

  }

  protected BSTNode<K, V> successor(BSTNode<K, V> xNode) {
    if (xNode == null) {
      return null;
    }
    if (xNode.right != null) {
      return minNode(xNode.right);
    }
    BSTNode<K, V> yNode = xNode.parent;
    while (yNode != null && xNode == yNode.right) {
      xNode = yNode;
      yNode = yNode.parent;
    }
    return yNode;
  }

  // 与找后继结点对称
  @Override
  public K predecessor(K x) {
    BSTNode<K, V> xNode = lookupNode(x);
    if (xNode == null) {
      return null;
    }
    if (xNode.left != null) {
      return maxNode(xNode.left).key;
    }
    BSTNode<K, V> yNode = xNode.parent;
    while (yNode != null && xNode.isLeftChild) {
      xNode = yNode;
      yNode = yNode.parent;
    }
    return yNode == null ? null : yNode.key;
  }

  @Override
  public boolean isBalance() {
    return !unBalance(root);
  }

  protected boolean unBalance(BSTNode g) {
    if (g == null) return false;
    int minus = getHeight(g.left) - getHeight(g.right);
    return Math.abs(minus) > 1
        || unBalance(g.right)
        || unBalance(g.left);
  }



  /**
   * 获取树的节点数
   * @return
   */
  @Override
  public int getSize() {
    return size;
  }

  @Override
  public int getHeight() {
    return getHeight(root);
  }

  protected int getHeight(BSTNode node) {
    if (node == null) return 0;
    int l = getHeight(node.left);
    int r = getHeight(node.right);
    return 1 + Math.max(l, r);
  }


  public List<List<BSTNode<K, V>>> levelOrder(BSTNode<K, V> x) {
    // int num=x.num;//累进的编号
    List<List<BSTNode<K, V>>> res = new ArrayList<>();
    Queue<BSTNode<K, V>> q = new LinkedList<>();
    q.add(x);
    BSTNode<K, V> last = x;
    BSTNode<K, V> nLast = null;
    List<BSTNode<K, V>> l = new ArrayList<>();
    res.add(l);
    while (!q.isEmpty()) {
      BSTNode<K, V> peek = q.peek();
      //把即将弹出的节点的子节点加入队列
      if (peek.left != null) {
        peek.left.num = peek.num * 2;
        q.add(peek.left);
        nLast = peek.left;
      }
      if (peek.right != null) {
        peek.right.num = peek.num * 2 + 1;
        q.add(peek.right);
        nLast = peek.right;
      }

      l.add(q.poll());//弹出,加入到当前层列表
      if (peek == last && !q.isEmpty()) {//如果现在弹出的节点是之前标记的最后节点，就要换列表
        l = new ArrayList<>();
        res.add(l);
        last = nLast;
      }
    }
    return res;
  }


  // 层次遍历
  @Override
  public List<List<BSTNode<K, V>>> levelOrder() {
    root.num = 1;
    return levelOrder(root);
  }

  // 按照格式打印
  @Override
  public String toString() {
    StringBuilder res = new StringBuilder();
    List<List<BSTNode<K, V>>> lists = levelOrder();
    int level = 1;
    int height = getHeight();
    for (List<BSTNode<K, V>> l :
        lists) {
      int gap = ex(2, height - level) - 1;//gap
      // printGap(gap);//打印左边margin
      int beginNum = ex(2, level - 1);
      for (BSTNode<K, V> node : l) {
        while (beginNum != node.num) {
          //打印gap
          for (int i = 0; i < 2 * gap; i++) {
            res.append(" ");
          }
          res.append("**");
          //打印gap
          for (int i = 0; i < 2 * gap; i++) {
            res.append(" ");
          }
          res.append("  ");
          beginNum++;
        }
        //打印gap
        for (int i = 0; i < 2 * gap; i++) {
          res.append(" ");
        }
        res.append(node.key);
        //打印gap
        for (int i = 0; i < 2 * gap; i++) {
          res.append(" ");
        }
        res.append("  ");

        beginNum++;
      }
      level++;
      res.append("\n");
    }
    return res.toString();
  }

  private void printGap(int margin) {
    for (int i = 0; i < margin; i++) {
      System.out.print(" ");
    }
  }

  private void printGap(int gap, String s, int gap1) {
    for (int i = 0; i < gap; i++) {
      System.out.print(" ");
    }
    System.out.printf("%2s", s);
    for (int i = 0; i < gap; i++) {
      System.out.print(" ");
    }

  }

    public static int ex(int a, int n) {
        if (n == 0)
            return 1;
        if (n == 1)
            return a;
        int temp = a; // a 的 1 次方
        int res = 1;
        int exponent = 1;
        while ((exponent << 1) < n) {
            temp = temp * temp;
            exponent = exponent << 1;
        }

        res *= ex(a, n - exponent);

        return res * temp;
    }
}