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en/docs/chapter_tree/binary_search_tree.md
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# 7.4   Binary search tree
As shown in Figure 7-16, a "binary search tree" satisfies the following conditions.
As shown in Figure 7-16, a <u>binary search tree</u> satisfies the following conditions.
1. For the root node, the value of all nodes in the left subtree < the value of the root node < the value of all nodes in the right subtree.
1. For the root node, the value of all nodes in the left subtree $<$ the value of the root node $<$ the value of all nodes in the right subtree.
2. The left and right subtrees of any node are also binary search trees, i.e., they satisfy condition `1.` as well.
![Binary search tree](binary_search_tree.assets/binary_search_tree.png){ class="animation-figure" }
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<p align="center"> Figure 7-20 &nbsp; Removing a node in a binary search tree (degree 1) </p>
When the degree of the node to be removed is $2$, we cannot remove it directly, but need to use a node to replace it. To maintain the property of the binary search tree "left subtree < root node < right subtree," **this node can be either the smallest node of the right subtree or the largest node of the left subtree**.
When the degree of the node to be removed is $2$, we cannot remove it directly, but need to use a node to replace it. To maintain the property of the binary search tree "left subtree $<$ root node $<$ right subtree," **this node can be either the smallest node of the right subtree or the largest node of the left subtree**.
Assuming we choose the smallest node of the right subtree (the next node in in-order traversal), then the removal operation proceeds as shown in Figure 7-21.
@@ -1705,7 +1705,7 @@ The operation of removing a node also uses $O(\log n)$ time, where finding the n
### 4. &nbsp; In-order traversal is ordered
As shown in Figure 7-22, the in-order traversal of a binary tree follows the "left $\rightarrow$ root $\rightarrow$ right" traversal order, and a binary search tree satisfies the size relationship "left child node < root node < right child node".
As shown in Figure 7-22, the in-order traversal of a binary tree follows the "left $\rightarrow$ root $\rightarrow$ right" traversal order, and a binary search tree satisfies the size relationship "left child node $<$ root node $<$ right child node".
This means that in-order traversal in a binary search tree always traverses the next smallest node first, thus deriving an important property: **The in-order traversal sequence of a binary search tree is ascending**.