AVL树

AVL树是二元搜索树，遵循每个节点的平衡条件。

• 平衡条件约束每个节点的子树高度相差不超过1。

AVL trees are binary search trees that obey a balance condition at each node.

• The balance condition constrains the height of the subtrees at each node to differ by no more than 1.

二叉树的高度

• 二叉树的高度是从根到叶的最长路径的长度。

• 具有一个节点的二叉树的高度为0。

• 按照惯例，空二进制树的高度为-1。

Height of Binary Trees

• The height of a binary tree is the length of the longest path from the root to a leaf.

• A binary tree with one node has height 0.

• An empty binary tree is has height -1 by convention.

AVL树的例子（Examples of AVL Trees）

非AVL树（A non-AVL Tree）

Root 7: Height: 0

Root 20: Height: 2

建立AVL树（Building AVL Trees）

• AVL树是通过从空的二叉树开始并一次添加一个元素来构建的。

• 对任何二叉搜索树进行添加，然后执行操作以恢复AVL平衡条件。

• AVL trees are built by starting with an empty binary tree and adding elements one at time.

• Additions are made as to any binary search tree, then an operation is executed to restore the AVL balance condition.

• 为了平衡自身，AVL树可以执行以下四种旋转

  • 左旋转

  • 右旋

  • 左右旋转（双旋转）

  •  左右旋转（双旋转）

• To balance itself, an AVL tree may perform the following four kinds of rotations

  • Left rotation

  • Right rotation

  • Left-Right rotation (Double Rotation) 

  • Right-Left rotation (Double Rotation)

LL不平衡

• 当N和N的左子节点K都是左重时，LL不平衡在节点N处发生。

LL imbalance

• An LL imbalance occurs at a node N with a left child K when N and K are both left-heavy.

• 如果节点的左子树具有比其右子树更高的高度，则节点是左重的。

• 通过在具有不平衡的节点处执行单个右旋转来校正LL不平衡

• A node is left-heavy if its left subtree has greater height then its right subtree.

• An LL imbalance is corrected by executing a single right rotation at the node with the imbalance

纠正LL不平衡（Correcting LL Imbalances）

RR不平衡

•当N和M都是右重时，在具有右子M的节点N处发生RR不平衡。

•RR不平衡是LL不平衡的镜像。

RR Imbalance

• An RR imbalance occurs at a node N with a right child M when N and M are both right-heavy.

• An RR imbalance is the mirror image of an LL imbalance.

纠正RR不平衡

•通过在具有不平衡的节点处执行单个左旋转来校正RR不平衡。

Correcting RR Imbalances

• An RR imbalance is corrected by executing a single left rotation at the node with the imbalance.

LR不平衡

• 当N为左重且K为右重时，LR不平衡发生在具有左子K的节点N处。

LR imbalance

• An LR imbalance occurs at a node N with a left child K when N is left-heavy and K is right- heavy.

纠正LR不平衡

• 通过在具有不平衡的节点处执行双LR旋转来校正LR不平衡。

Correcting An LR Imbalance

• An LR imbalance is corrected by executing a double LR rotation at the node with the imbalance.

•通过在具有不平衡的节点处执行双LR旋转来校正LR不平衡。

• An LR imbalance is corrected by executing a double LR rotation at the node with the imbalance.

RL不平衡（RL imbalance）

•当N为右重且K为左重时，在具有右子K的节点N处发生RL不平衡。

• An RL imbalance occurs at a node N with a right child K when N is right-heavy and K is left- heavy.

纠正RL不平衡•左右旋转

Correcting RL imbalance • Right-Left Rotation