JavaScript 数据结构与算法 之 树

树

数据结构

• 相关术语

  • 位于树顶部的节点叫作根节点
  • 至少有一个子节点的节点称为内部节点
  • 没有子元素的节点称为外部节点或叶节点
  • 子树由节点和它的后代构成
  • 节点的深度取决于它的祖先节点的数量
  • 树的高度取决于所有节点深度的最大值。一棵树也可以被分解成层级，根节点在第 0 层。

二叉树

二叉树中的节点最多只能有两个子节点：一个是左侧子节点，另一个是右侧子节点。二叉搜索树（BST）是二叉树的一种，但是只允许在左侧节点存储（比父节点）小的值，在右侧节点存储（比父节点）大的值。

class Node {
  constructor() {
    this.key = key;
    this.left = null;
    this.right = null;
  }
}
class BinarySearchTree {
  constructor(compareFn = defaultCompare) {
    this.compareFn = compareFn;
    this.root = null;
  }
  insert(key) {
    if (this.root == null) {
      this.root = new Node(key);
    } else {
      this.insertNode(this.root, key);
    }
  }
  insertNode(node, key) {
    if (compareFn(key, node.key) === Compare.LESS_THAN) {
      if (node.left != null) {
        node.left = new Node(key);
      } else {
        this.insertNode(node.left, key);
      }
    } else {
      if (node.right != null) {
        node.right = new Node(key);
      } else {
        this.insertNode(node.right, key);
      }
    }
  }
  // 中序遍历
  inOrderTraverse(cb) {
    this.inOrderTraverseNode(this.root, cb);
  }
  inOrderTraverseNode(node, cb) {
    if (node != null) {
      this.inOrderTraverseNode(node.left, cb);
      cb(node.key);
      this.inOrderTraverseNode(node.right, cb);
    }
  }
  // 先序遍历
  preOrderTraverse(cb) {
    this.preOrderTraverseNode(this.root, cb);
  }
  preOrderTraverseNode(node, cb) {
    if (node != null) {
      cb(node.key);
      this.preOrderTraverseNode(node.left, cb);
      this.preOrderTraverseNode(node.right, cb);
    }
  }
  // 后续遍历
  postOrderTraverse(cb) {
    this.postOrderTraverseNode(this.root, cb);
  }
  postOrderTraverseNode(node, cb) {
    if (node != null) {
      this.postOrderTraverseNode(node.left, cb);
      this.postOrderTraverseNode(node.right, cb);
      cb(node.key);
    }
  }
  // 搜索最小值
  min() {
    return this.miniNode(this.root);
  }
  miniNode(node) {
    let current = node;
    while(current != null && node.left != null) {
      current = current.left;
    }
    return current;
  }
  // 最大值
  max() {
    return this.maxNode(this.root);
  }
  maxNode(node) {
    let current = node;
    while(current != null && node.right != null) {
      current = current.right
    }
    return current;
  }
  // 搜索特定的值
  search(key) {
    return this.searchNode(this.root, key);
  }
  searchNode(node, key) {
    if (node == null) {
      return false;
    }
    if (this.compareFn(key, node.key) === Compare.LESS_THAN) {
      return this.searchNode(node.left, key);
    } else if (this.compareFn(key, node.key) === Compare.BIGGER_THAN) {
      return this.searchNode(node.right, key);
    } else {
      return true;
    }
  }
  
  remove(key) {
    this.root = this.removeNode(this.root, key);
  }
  removeNode(node, key) {
    if (node == null) {
      return null;
    }
    if (this.compareFn(key, node.key) === Compare.LESS_THAN) {
      node.left = this.removeNode(node.left, key);
      return node;
    } else if (this.compareFn(key, node.key) === Compare.BIGGER_THAN) {
      node.right = this.removeNode(node.right, key);
      return node;
    } else {
      // 无子节点
      if (node.left == null && node.right == null) {
        node = null;
        return node;
      }
      // 有一个子节点
      if (node.left == null) {
        node = node.right;
        return node;
      } else if (node.right == null) {
        node = node.left;
        return node;
      }
      // 有两个子节点
      const aux = this.miniNode(node.right);
      node.key = aux.key;
      node.right = this.removeNode(node.right, aux.key);
      return node;
    }
  }
}

自平衡树

Adelson-Velskii-Landi 树（ AVL 树）是一种自平衡二叉搜索树，意思是任何一个节点左右两侧子树的高度之差最多为 1。

在 AVL 树中，需要对每个节点计算右子树高度（ hr）和左子树高度（ hl）之间的差值，该值（ hr－ hl）应为 0、 1 或1。如果结果不是这三个值之一，则需要平衡该 AVL 树。这就是平衡因子的概念。

• 平衡操作——AVL 旋转

  • 左-左（LL）：向右的单旋转

  
  

  • 右-右（RR）：向左的单旋转

  
  

  • 左-右（LR）：向右的双旋转（先 LL 旋转，再 RR 旋转）

  
  

  • 右-左（RL）：向左的双旋转（先 RR 旋转，再 LL 旋转）

  

const BalanceFactor = {
  UNBALANCED_RIGHT: 1,
  SLIGHTLY_UNBALANCED_RIGHT: 2,
  BALANCED: 3,
  SLIGHTLY_UNBALANCED_LEFT: 4,
  UNBALANCED_LEFT: 5
};
class AVLTree extends BinarySearchTree {
  constructor(compareFn = defaultCompare) {
    super(compareFn);
    this.compareFn = compareFn;
    this.root = null;
  }
  // 计算节点高度
  getNodeHeight(node) {
    if (node == null) {
      return -1;
    }
    return Math.max(this.getNodeHeight(node.left), this.getNodeHeight(node.right)) + 1;
  }
  // 计算平衡因子
  getBalanceFactor(node) {
    const heightDifference = this.getNodeHeight(node.left) - this.getNodeHeight(node.right);
    switch(heightDifference) {
      case -2:
        return BalanceFactor.UNBALANCED_RIGHT;
      case -1:
        return BalanceFactor.SLIGHTLY_UNBALANCED_RIGHT;
      case 1:
        return BalanceFactor.SLIGHTLY_UNBALANCED_LEFT;
      case 2:
        return BalanceFactor.UNBALANCED_LEFT;
      default:
        return BalanceFactor.BALANCED;
    }
  }
  rotationLL(node) {
    const tmp = node.left;
    node.left = tmp.right;
    tmp.right = node;
    return tmp;
  }
  rotationRR(node) {
    const tmp = node.right;
    node.right = tmp.left;
    tmp.left = node;
    return tmp;
  }
  rotationLR(node) {
    node.left = this.rotationRR(node.left);
    return this.rotationLL(node);
  }
  rotationRL(node) {
    node.right = this.rotationLL(node.right);
    return this.rotationRR(node);
  }

  insert(key) {
    this.root = this.insertNode(this.root, key);
  }
  insertNode(node, key) {
    // 先像 BST 中一样插入
    if (node == null) {
      return new Node(key);
    } else if (this.compareFn(key, node.key) === Compare.LESS_THAN) {
      node.left = this.insertNode(node.left, key);
    } else if (this.compareFn(key, node.key) === Compare.BIGGER_THAN) {
      node.right = this.insertNode(node.right, key);
    } else { // 重复的键
      return node;
    }
    // 如果结果不平衡就进行调整
    const balanceFactor = this.getBalanceFactor(node);
    if balanceFactor === BalanceFactor.UNBALANCED_LEFT) {
      if (this.compareFn(key, node.left.key) === Compare.LESS_THAN) {
        node = this.rotationLL(node);
      } else {
        return this.rotationLR(node);
      }
    }
    if (balanceFactor === BalanceFactor.UNBALANCED_RIGHT) {
      if (this.compareFn(key, node.right.key) === Compare.BIGGER_THAN) {
        node = this.rotationRR(node);
      } else {
        return this.rotationRL(node);
      }
    }
    return node;
  }

  removeNode(node, key) {
    node = super.removeNode(node, key);
    if (node == null) {
      return node;
    }
    const balanceFactor = this.getBalanceFactor(node);
    if (balanceFactor === BalanceFactor.UNBALANCED_LEFT) {
      const balanceFactorLeft = this.getBalanceFactor(node.left);
      if (balanceFactorLeft === BalanceFactor.BALANCED ||
        balanceFactorLeft === BalanceFactor.SLIGHTLY_UNBALANCED_LEFT) {
        return this.rotationLL(node);
      }
      if (balanceFactorLeft === BalanceFactor.SLIGHTLY_UNBALANCED_RIGHT) {
        return this.rotationLR(node);
      }
    }
    if (balanceFactor === BalanceFactor.UNBALANCED_RIGHT) {
      const balanceFactorRight = this.getBalanceFactor(node.right);
      if (balanceFactorRight === BalanceFactor.BALANCED ||
        balanceFactorRight === BalanceFactor.SLIGHTLY_UNBALANCED_RIGHT) {
        return this.rotationRR(node);
      }
      if (balanceFactorRight === BalanceFactor.SLIGHTLY_UNBALANCED_LEFT) {
        return this.rotationRL(node);
      }
    }
    return node;
  }
}

红黑树

红黑树也是一个自平衡二叉搜索树。如果插入和删除频率较低，AVL 树比红黑树更好。

• 红黑树定义规则

  • 每个节点不是红的就是黑的
  • 树的根节点是黑的
  • 所有叶节点都是黑的（用 NULL 引用表示的节点）
  • 如果一个节点是红的，那么它的两个子节点都是黑的
  • 不能有两个相邻的红节点，一个红节点不能有红的父节点或子节点
  • 从给定的节点到它的后代节点（ NULL 叶节点）的所有路径包含相同数量的黑色节点

• 旋转和颜色变换规则

  • 变颜色：当前节点的父亲是红色，且它的祖父节点的另一个子节点（叔叔节点）也是红色

    1. 把父节点设为黑色
    2. 把叔叔节点也设为黑色
    3. 把祖父节点设为红色
    4. 把指针定义到祖父节点设为当前要操作的，分析节点变换规则
  • 左旋：当前父亲节点是红色，叔叔节点是黑色，且当前节点是右子树，左旋以父节点作为左旋

  • 右旋：当前父节点是红色，叔叔是黑色的时候，且当前的节点是左子树

    1. 把父节点变为黑色
    2. 将祖父节点变为红色
    3. 以祖父节点旋转

class RedBlackNode extends Node {
  constructor(key) {
    super(key);
    this.key = key;
    this.color = Colors.RED;
    this.parent = null;
  }
  isRed() {
    return this.color === Colors.RED;
  }
}
class RedBlackTree extends BinarySearchTree {
  constructor(compareFn = defaultCompare) {
    super(compareFn);
    this.compareFn = compareFn;
    this.root = null;
  }
  insert(key: T) {
    if (this.root == null) {
      this.root = new RedBlackNode(key);
      this.root.color = Colors.BLACK;
    } else {
      const newNode = this.insertNode(this.root, key);
      this.fixTreeProperties(newNode);
    }
  }
  insertNode(node, key) {
    if (this.compareFn(key, node.key) === Compare.LESS_THAN) {
      if (node.left == null) {
        node.left = new RedBlackNode(key);
        node.left.parent = node;
        return node.left;
      } else {
        return this.insertNode(node.left, key);
      }
    } else if (node.right == null) {
      node.right = new RedBlackNode(key);
      node.right.parent = node;
      return node.right;
    } else {
      return this.insertNode(node.right, key);
    }
  }
  fixTreeProperties(node) {
    while (node && node.parent && node.parent.isRed()
      && node.color !== Colors.BLACK) {
      let parent = node.parent;
      const grandParent = parent.parent;
      // 父节点是左侧子节点
      if (grandParent && grandParent.left === parent) {
        const uncle = grandParent.right;
        // 叔叔节点也是红色——只需要重新填色
        if (uncle && uncle.color === Colors.RED) {
          grandParent.color = Colors.RED;
          parent.color = Colors.BLACK;
          uncle.color = Colors.BLACK;
          node = grandParent;
        } else {
          // 节点是右侧子节点——左旋转
          if (node === parent.right) {
            this.rotationRR(parent);
            node = parent;
            parent = node.parent;
          }
          // 节点是左侧子节点——右旋转
          this.rotationLL(grandParent);
          parent.color = Colors.BLACK;
          grandParent.color = Colors.RED;
          node = parent;
        }
      } else {
        // 父节点是右侧子节点
        const uncle = grandParent.left;
        // 叔叔节点是红色——只需要重新填色
        if (uncle && uncle.color === Colors.RED) {
          grandParent.color = Colors.RED;
          parent.color = Colors.BLACK;
          uncle.color = Colors.BLACK;
          node = grandParent;
        } else {
          // 节点是左侧子节点——右旋转
          if (node === parent.left) {
            this.rotationLL(parent);
            node = parent;
            parent = node.parent;
          }
          // 节点是右侧子节点——左旋转
          this.rotationRR(grandParent);
          parent.color = Colors.BLACK;
          grandParent.color = Colors.RED;
          node = parent;
        }
      }
    }
    this.root.color = Colors.BLACK;
  }
  rotationLL(node) {
    const tmp = node.left;
    node.left = tmp.right;
    if (tmp.right && tmp.right.key) {
      tmp.right.parent = node;
    }
    tmp.parent = node.parent;
    if (!node.parent) {
      this.root = tmp;
    } else {
      if (node === node.parent.left) {
        node.parent.left = tmp;
      } else {
        node.parent.right = tmp;
      }
    }
    tmp.right = node;
    node.parent = tmp;
  }
  rotationRR(node) {
    const tmp = node.right;
    node.right = tmp.left;
    if (tmp.left && tmp.left.key) {
      tmp.left.parent = node;
    }
    tmp.parent = node.parent;
    if (!node.parent) {
      this.root = tmp;
    }
    else {
      if (node === node.parent.left) {
        node.parent.left = tmp;
      }
      else {
        node.parent.right = tmp;
      }
    }
    tmp.left = node;
    node.parent = tmp;
  }
}