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en/docs/chapter_tree/avl_tree.md
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In the "Binary Search Tree" section, we mentioned that after multiple insertions and removals, a binary search tree might degrade to a linked list. In such cases, the time complexity of all operations degrades from $O(\log n)$ to $O(n)$.
As shown in the Figure 7-24 , after two node removal operations, this binary search tree will degrade into a linked list.
As shown in Figure 7-24, after two node removal operations, this binary search tree will degrade into a linked list.
![Degradation of an AVL tree after removing nodes](avl_tree.assets/avltree_degradation_from_removing_node.png){ class="animation-figure" }
<p align="center"> Figure 7-24 &nbsp; Degradation of an AVL tree after removing nodes </p>
For example, in the perfect binary tree shown in the Figure 7-25 , after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.
For example, in the perfect binary tree shown in Figure 7-25, after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.
![Degradation of an AVL tree after inserting nodes](avl_tree.assets/avltree_degradation_from_inserting_node.png){ class="animation-figure" }
@@ -682,7 +682,7 @@ We call nodes with an absolute balance factor $> 1$ "unbalanced nodes". Dependin
### 1. &nbsp; Right rotation
As shown in the Figure 7-26 , the first unbalanced node from the bottom up in the binary tree is "node 3". Focusing on the subtree with this unbalanced node as the root, denoted as `node`, and its left child as `child`, perform a "right rotation". After the right rotation, the subtree is balanced again while still maintaining the properties of a binary search tree.
As shown in Figure 7-26, the first unbalanced node from the bottom up in the binary tree is "node 3". Focusing on the subtree with this unbalanced node as the root, denoted as `node`, and its left child as `child`, perform a "right rotation". After the right rotation, the subtree is balanced again while still maintaining the properties of a binary search tree.
=== "<1>"
![Steps of right rotation](avl_tree.assets/avltree_right_rotate_step1.png){ class="animation-figure" }
@@ -698,7 +698,7 @@ As shown in the Figure 7-26 , the first unbalanced node from the bottom up in th
<p align="center"> Figure 7-26 &nbsp; Steps of right rotation </p>
As shown in the Figure 7-27 , when the `child` node has a right child (denoted as `grand_child`), a step needs to be added in the right rotation: set `grand_child` as the left child of `node`.
As shown in Figure 7-27, when the `child` node has a right child (denoted as `grand_child`), a step needs to be added in the right rotation: set `grand_child` as the left child of `node`.
![Right rotation with grand_child](avl_tree.assets/avltree_right_rotate_with_grandchild.png){ class="animation-figure" }
@@ -965,13 +965,13 @@ As shown in the Figure 7-27 , when the `child` node has a right child (denoted a
### 2. &nbsp; Left rotation
Correspondingly, if considering the "mirror" of the above unbalanced binary tree, the "left rotation" operation shown in the Figure 7-28 needs to be performed.
Correspondingly, if considering the "mirror" of the above unbalanced binary tree, the "left rotation" operation shown in Figure 7-28 needs to be performed.
![Left rotation operation](avl_tree.assets/avltree_left_rotate.png){ class="animation-figure" }
<p align="center"> Figure 7-28 &nbsp; Left rotation operation </p>
Similarly, as shown in the Figure 7-29 , when the `child` node has a left child (denoted as `grand_child`), a step needs to be added in the left rotation: set `grand_child` as the right child of `node`.
Similarly, as shown in Figure 7-29, when the `child` node has a left child (denoted as `grand_child`), a step needs to be added in the left rotation: set `grand_child` as the right child of `node`.
![Left rotation with grand_child](avl_tree.assets/avltree_left_rotate_with_grandchild.png){ class="animation-figure" }
@@ -1238,7 +1238,7 @@ It can be observed that **the right and left rotation operations are logically s
### 3. &nbsp; Right-left rotation
For the unbalanced node 3 shown in the Figure 7-30 , using either left or right rotation alone cannot restore balance to the subtree. In this case, a "left rotation" needs to be performed on `child` first, followed by a "right rotation" on `node`.
For the unbalanced node 3 shown in Figure 7-30, using either left or right rotation alone cannot restore balance to the subtree. In this case, a "left rotation" needs to be performed on `child` first, followed by a "right rotation" on `node`.
![Right-left rotation](avl_tree.assets/avltree_left_right_rotate.png){ class="animation-figure" }
@@ -1246,7 +1246,7 @@ For the unbalanced node 3 shown in the Figure 7-30 , using either left or right
### 4. &nbsp; Left-right rotation
As shown in the Figure 7-31 , for the mirror case of the above unbalanced binary tree, a "right rotation" needs to be performed on `child` first, followed by a "left rotation" on `node`.
As shown in Figure 7-31, for the mirror case of the above unbalanced binary tree, a "right rotation" needs to be performed on `child` first, followed by a "left rotation" on `node`.
![Left-right rotation](avl_tree.assets/avltree_right_left_rotate.png){ class="animation-figure" }
@@ -1254,13 +1254,13 @@ As shown in the Figure 7-31 , for the mirror case of the above unbalanced binary
### 5. &nbsp; Choice of rotation
The four kinds of imbalances shown in the Figure 7-32 correspond to the cases described above, respectively requiring right rotation, left-right rotation, right-left rotation, and left rotation.
The four kinds of imbalances shown in Figure 7-32 correspond to the cases described above, respectively requiring right rotation, left-right rotation, right-left rotation, and left rotation.
![The four rotation cases of AVL tree](avl_tree.assets/avltree_rotation_cases.png){ class="animation-figure" }
<p align="center"> Figure 7-32 &nbsp; The four rotation cases of AVL tree </p>
As shown in the Table 7-3 , we determine which of the above cases an unbalanced node belongs to by judging the sign of the balance factor of the unbalanced node and its higher-side child's balance factor.
As shown in Table 7-3, we determine which of the above cases an unbalanced node belongs to by judging the sign of the balance factor of the unbalanced node and its higher-side child's balance factor.
<p align="center"> Table 7-3 &nbsp; Conditions for Choosing Among the Four Rotation Cases </p>