diff --git a/en/docs/chapter_tree/binary_tree_traversal.md b/en/docs/chapter_tree/binary_tree_traversal.md index ab29d041c..08c104605 100755 --- a/en/docs/chapter_tree/binary_tree_traversal.md +++ b/en/docs/chapter_tree/binary_tree_traversal.md @@ -1,14 +1,14 @@ # Binary tree traversal -From the perspective of physical structure, a tree is a data structure based on linked lists, hence its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms to achieve. +From a physical structure perspective, a tree is a data structure based on linked lists. Hence, its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms. -Common traversal methods for binary trees include level-order traversal, pre-order traversal, in-order traversal, and post-order traversal, among others. +The common traversal methods for binary trees include level-order traversal, pre-order traversal, in-order traversal, and post-order traversal. ## Level-order traversal -As shown in the figure below, level-order traversal traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order. +As shown in the figure below, level-order traversal traverses the binary tree from top to bottom, layer by layer. Within each level, it visits nodes from left to right. -Level-order traversal essentially belongs to breadth-first traversal, also known as breadth-first search (BFS), which embodies a "circumferentially outward expanding" layer-by-layer traversal method. +Level-order traversal is essentially a type of breadth-first traversal, also known as breadth-first search (BFS), which embodies a "circumferentially outward expanding" layer-by-layer traversal method. ![Level-order traversal of a binary tree](binary_tree_traversal.assets/binary_tree_bfs.png) @@ -22,14 +22,14 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q ### Complexity analysis -- **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time, where $n$ is the number of nodes. -- **Space complexity is $O(n)$**: In the worst case, i.e., a full binary tree, before traversing to the lowest level, the queue can contain at most $(n + 1) / 2$ nodes at the same time, occupying $O(n)$ space. +- **Time complexity is $O(n)$**: All nodes are visited once, taking $O(n)$ time, where $n$ is the number of nodes. +- **Space complexity is $O(n)$**: In the worst case, i.e., a full binary tree, before traversing to the bottom level, the queue can contain at most $(n + 1) / 2$ nodes simultaneously, occupying $O(n)$ space. ## Preorder, in-order, and post-order traversal Correspondingly, pre-order, in-order, and post-order traversal all belong to depth-first traversal, also known as depth-first search (DFS), which embodies a "proceed to the end first, then backtrack and continue" traversal method. -The figure below shows the working principle of performing a depth-first traversal on a binary tree. **Depth-first traversal is like walking around the perimeter of the entire binary tree**, encountering three positions at each node, corresponding to pre-order traversal, in-order traversal, and post-order traversal. +The figure below shows the working principle of performing a depth-first traversal on a binary tree. **Depth-first traversal is like "walking" around the entire binary tree**, encountering three positions at each node, corresponding to pre-order, in-order, and post-order traversal. ![Preorder, in-order, and post-order traversal of a binary search tree](binary_tree_traversal.assets/binary_tree_dfs.png) @@ -86,4 +86,4 @@ The figure below shows the recursive process of pre-order traversal of a binary ### Complexity analysis - **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time. -- **Space complexity is $O(n)$**: In the worst case, i.e., the tree degrades into a linked list, the recursion depth reaches $n$, the system occupies $O(n)$ stack frame space. +- **Space complexity is $O(n)$**: In the worst case, i.e., the tree degenerates into a linked list, the recursion depth reaches $n$, the system occupies $O(n)$ stack frame space.