Revisit the English version (#1835)

* Review the English version using Claude-4.5.

* Update mkdocs.yml

* Align the section titles.

* Bug fixes

en/docs/chapter_heap/summary.md

@@ -2,16 +2,16 @@
 
 ### Key review
 
-- A heap is a complete binary tree that can be categorized as either a max heap or a min heap based on its building property, where the top element of a max heap is the largest and the top element of a min heap is the smallest.
-- A priority queue is defined as a queue with dequeue priority, usually implemented using a heap.
-- Common operations of a heap and their corresponding time complexities include: element insertion into the heap $O(\log n)$, removing the top element from the heap $O(\log n)$, and accessing the top element of the heap $O(1)$.
-- A complete binary tree is well-suited to be represented by an array, thus heaps are commonly stored using arrays.
-- Heapify operations are used to maintain the properties of the heap and are used in both heap insertion and removal operations.
-- The time complexity of building a heap given an input of $n$ elements can be optimized to $O(n)$, which is highly efficient.
+- A heap is a complete binary tree that can be categorized as a max heap or min heap based on its property. The heap top element of a max heap (min heap) is the largest (smallest).
+- A priority queue is defined as a queue with priority sorting, typically implemented using heaps.
+- Common heap operations and their corresponding time complexities include: element insertion $O(\log n)$, heap top element removal $O(\log n)$, and accessing the heap top element $O(1)$.
+- Complete binary trees are well-suited for array representation, so we typically use arrays to store heaps.
+- Heapify operations are used to maintain the heap property and are employed in both element insertion and removal operations.
+- The time complexity of building a heap with $n$ input elements can be optimized to $O(n)$, which is highly efficient.
 - Top-k is a classic algorithm problem that can be efficiently solved using the heap data structure, with a time complexity of $O(n \log k)$.
 
 ### Q & A
 
-**Q**: Is the "heap" in data structures the same concept as the "heap" in memory management?
+**Q**: Are the "heap" in data structures and the "heap" in memory management the same concept?
 
-The two are not the same concept, even though they are both referred to as "heap". The heap in computer system memory is part of dynamic memory allocation, where the program can use it to store data during execution. The program can request a certain amount of heap memory to store complex structures like objects and arrays. When the allocated data is no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, the management and usage of heap memory demands more caution, as improper use may lead to memory leaks and dangling pointers.
+The two are not the same concept; they just happen to share the name "heap." The heap in computer system memory is part of dynamic memory allocation, where programs can use it to store data during runtime. Programs can request a certain amount of heap memory to store complex structures such as objects and arrays. When this data is no longer needed, the program needs to release this memory to prevent memory leaks. Compared to stack memory, heap memory management and usage require more caution, as improper use can lead to issues such as memory leaks and dangling pointers.