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Bug fixes and improvements (#1348)

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* Add "reference" for EN version. Bug fixes.

* Unify the figure reference as "the figure below" and "the figure above".
Bug fixes.

* Format the EN markdown files.

* Replace "" with <u></u> for EN version and bug fixes

* Fix biary_tree_dfs.png

* Fix biary_tree_dfs.png

* Fix zh-hant/biary_tree_dfs.png

* Fix heap_sort_step1.png

* Sync zh and zh-hant versions.

* Bug fixes

* Fix EN figures

* Bug fixes

* Fix the figure labels for EN version
This commit is contained in:
Yudong Jin
2024-05-06 14:44:48 +08:00
committed by GitHub
parent 8e60d12151
commit c4a7966882
99 changed files with 615 additions and 259 deletions
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en/docs/chapter_tree/binary_search_tree.md
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@@ -1,8 +1,8 @@
# Binary search tree
As shown in the figure below, a "binary search tree" satisfies the following conditions.
As shown in the figure below, 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)
@@ -69,7 +69,7 @@ As shown in the figure below, when the degree of the node to be removed is $1$,
![Removing a node in a binary search tree (degree 1)](binary_search_tree.assets/bst_remove_case2.png)
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 the figure below.
@@ -96,7 +96,7 @@ The operation of removing a node also uses $O(\log n)$ time, where finding the n
### In-order traversal is ordered
As shown in the figure below, 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 the figure below, 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**.