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Translate all code to English (#1836)

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* Review the EN heading format.

* Fix pythontutor headings.

* Fix pythontutor headings.

* bug fixes

* Fix headings in **/summary.md

* Revisit the CN-to-EN translation for Python code using Claude-4.5

* Revisit the CN-to-EN translation for Java code using Claude-4.5

* Revisit the CN-to-EN translation for Cpp code using Claude-4.5.

* Fix the dictionary.

* Fix cpp code translation for the multipart strings.

* Translate Go code to English.

* Update workflows to test EN code.

* Add EN translation for C.

* Add EN translation for CSharp.

* Add EN translation for Swift.

* Trigger the CI check.

* Revert.

* Update en/hash_map.md

* Add the EN version of Dart code.

* Add the EN version of Kotlin code.

* Add missing code files.

* Add the EN version of JavaScript code.

* Add the EN version of TypeScript code.

* Fix the workflows.

* Add the EN version of Ruby code.

* Add the EN version of Rust code.

* Update the CI check for the English version  code.

* Update Python CI check.

* Fix cmakelists for en/C code.

* Fix Ruby comments
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Yudong Jin
2025-12-31 07:44:52 +08:00
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en/docs/chapter_tree/array_representation_of_tree.md
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# Array representation of binary trees
# Array Representation of Binary Trees
Under the linked list representation, the storage unit of a binary tree is a node `TreeNode`, and nodes are connected by pointers. The previous section introduced the basic operations of binary trees under the linked list representation.
So, can we use an array to represent a binary tree? The answer is yes.
## Representing perfect binary trees
## Representing Perfect Binary Trees
Let's analyze a simple case first. Given a perfect binary tree, we store all nodes in an array according to the order of level-order traversal, where each node corresponds to a unique array index.
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**The mapping formula plays a role similar to the node references (pointers) in linked lists**. Given any node in the array, we can access its left (right) child node using the mapping formula.
## Representing any binary tree
## Representing Any Binary Tree
Perfect binary trees are a special case; in the middle levels of a binary tree, there are typically many `None` values. Since the level-order traversal sequence does not include these `None` values, we cannot infer the number and distribution of `None` values based on this sequence alone. **This means multiple binary tree structures can correspond to the same level-order traversal sequence**.
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[file]{array_binary_tree}-[class]{array_binary_tree}-[func]{}
```
## Advantages and limitations
## Advantages and Limitations
The array representation of binary trees has the following advantages: