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en/docs/chapter_computational_complexity/iteration_and_recursion.md
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@@ -1220,7 +1220,7 @@ Observe the following code, where simply calling the function `recur(n)` can com
<div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20%23%20%E9%80%92%EF%BC%9A%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%0A%20%20%20%20res%20%3D%20recur%28n%20-%201%29%0A%20%20%20%20%23%20%E5%BD%92%EF%BC%9A%E8%BF%94%E5%9B%9E%E7%BB%93%E6%9E%9C%0A%20%20%20%20return%20n%20%2B%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20recur%28n%29%0A%20%20%20%20print%28f%22%5Cn%E9%80%92%E5%BD%92%E5%87%BD%E6%95%B0%E7%9A%84%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20%23%20%E9%80%92%EF%BC%9A%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%0A%20%20%20%20res%20%3D%20recur%28n%20-%201%29%0A%20%20%20%20%23%20%E5%BD%92%EF%BC%9A%E8%BF%94%E5%9B%9E%E7%BB%93%E6%9E%9C%0A%20%20%20%20return%20n%20%2B%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20recur%28n%29%0A%20%20%20%20print%28f%22%5Cn%E9%80%92%E5%BD%92%E5%87%BD%E6%95%B0%E7%9A%84%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
The Figure 2-3 shows the recursive process of this function.
Figure 2-3 shows the recursive process of this function.
![Recursive process of the sum function](iteration_and_recursion.assets/recursion_sum.png){ class="animation-figure" }
@@ -1243,7 +1243,7 @@ Every time a recursive function calls itself, the system allocates memory for th
- The function's context data is stored in a memory area called "stack frame space" and is only released after the function returns. Therefore, **recursion generally consumes more memory space than iteration**.
- Recursive calls introduce additional overhead. **Hence, recursion is usually less time-efficient than loops.**
As shown in the Figure 2-4 , there are $n$ unreturned recursive functions before triggering the termination condition, indicating a **recursion depth of $n$**.
As shown in Figure 2-4, there are $n$ unreturned recursive functions before triggering the termination condition, indicating a **recursion depth of $n$**.
![Recursion call depth](iteration_and_recursion.assets/recursion_sum_depth.png){ class="animation-figure" }
@@ -1447,7 +1447,7 @@ For example, in calculating $1 + 2 + \dots + n$, we can make the result variable
<div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20tail_recur%28n,%20res%29%3A%0A%20%20%20%20%22%22%22%E5%B0%BE%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20res%0A%20%20%20%20%23%20%E5%B0%BE%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%0A%20%20%20%20return%20tail_recur%28n%20-%201,%20res%20%2B%20n%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20tail_recur%28n,%200%29%0A%20%20%20%20print%28f%22%5Cn%E5%B0%BE%E9%80%92%E5%BD%92%E5%87%BD%E6%95%B0%E7%9A%84%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20tail_recur%28n,%20res%29%3A%0A%20%20%20%20%22%22%22%E5%B0%BE%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E7%BB%88%E6%AD%A2%E6%9D%A1%E4%BB%B6%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20res%0A%20%20%20%20%23%20%E5%B0%BE%E9%80%92%E5%BD%92%E8%B0%83%E7%94%A8%0A%20%20%20%20return%20tail_recur%28n%20-%201,%20res%20%2B%20n%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20tail_recur%28n,%200%29%0A%20%20%20%20print%28f%22%5Cn%E5%B0%BE%E9%80%92%E5%BD%92%E5%87%BD%E6%95%B0%E7%9A%84%E6%B1%82%E5%92%8C%E7%BB%93%E6%9E%9C%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
The execution process of tail recursion is shown in the following figure. Comparing regular recursion and tail recursion, the point of the summation operation is different.
The execution process of tail recursion is shown in Figure 2-5. Comparing regular recursion and tail recursion, the point of the summation operation is different.
- **Regular recursion**: The summation operation occurs during the "returning" phase, requiring another summation after each layer returns.
- **Tail recursion**: The summation operation occurs during the "calling" phase, and the "returning" phase only involves returning through each layer.