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<a href=".." >递归函数</a>
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<h1 id="4-递归函数">4. 递归函数</h1>
<h2 id="41-递归函数的定义">4.1 递归函数的定义</h2>
<p>递归函数即函数在内部调用函数本身（自己调用自己）。</p>
<p>递归的关键点：</p>
<ul>
<li>递归结束的临界值，通常使用if语句来判断临界值</li>
<li>递归体的递归逻辑</li>
</ul>
<p>例如，以下是fact(n) = n! = 1 x 2 x 3 x ... x (n-1) x n = (n-1)! x n = fact(n-1) x n 的递归函数。</p>
<pre class="language-"><code class="lang-python"><span class="token comment">#递归函数的定义</span>
<span class="token keyword">def</span> <span class="token function">fact</span><span class="token punctuation">(</span>n<span class="token punctuation">)</span><span class="token punctuation">:</span>
<span class="token keyword">if</span> n<span class="token operator">==</span><span class="token number">1</span><span class="token punctuation">:</span>
<span class="token keyword">return</span> <span class="token number">1</span>
<span class="token keyword">return</span> n <span class="token operator">*</span> fact<span class="token punctuation">(</span>n <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">)</span>
<span class="token comment">#计算5的阶乘，递归函数调用细节</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> fact<span class="token punctuation">(</span><span class="token number">5</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> fact<span class="token punctuation">(</span><span class="token number">4</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">4</span> <span class="token operator">*</span> fact<span class="token punctuation">(</span><span class="token number">3</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">4</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">3</span> <span class="token operator">*</span> fact<span class="token punctuation">(</span><span class="token number">2</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">4</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">3</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> fact<span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">4</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">3</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> <span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">4</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">3</span> <span class="token operator">*</span> <span class="token number">2</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token punctuation">(</span><span class="token number">4</span> <span class="token operator">*</span> <span class="token number">6</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">5</span> <span class="token operator">*</span> <span class="token number">24</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">120</span>
</code></pre>
<h2 id="42-栈溢出问题">4.2 栈溢出问题</h2>
<p>​ 在计算机中，函数调用是通过栈（stack）这种数据结构实现的，每当进入一个函数调用，栈就会加一层栈帧，每当函数返回，栈就会减一层栈帧。由于栈的大小不是无限的，所以，递归调用的次数过多，会导致栈溢出。</p>
<p>例如：</p>
<pre class="language-"><code class="lang-python"><span class="token operator">>></span><span class="token operator">></span> fact<span class="token punctuation">(</span><span class="token number">1000</span><span class="token punctuation">)</span>
Traceback <span class="token punctuation">(</span>most recent call last<span class="token punctuation">)</span><span class="token punctuation">:</span>
File <span class="token string">"<stdin>"</span><span class="token punctuation">,</span> line <span class="token number">1</span><span class="token punctuation">,</span> <span class="token keyword">in</span> <span class="token operator"><</span>module<span class="token operator">></span>
File <span class="token string">"<stdin>"</span><span class="token punctuation">,</span> line <span class="token number">4</span><span class="token punctuation">,</span> <span class="token keyword">in</span> fact
<span class="token punctuation">.</span><span class="token punctuation">.</span><span class="token punctuation">.</span>
File <span class="token string">"<stdin>"</span><span class="token punctuation">,</span> line <span class="token number">4</span><span class="token punctuation">,</span> <span class="token keyword">in</span> fact
RuntimeError<span class="token punctuation">:</span> maximum recursion depth exceeded <span class="token keyword">in</span> comparison
</code></pre>
<h2 id="43-尾递归">4.3 尾递归</h2>
<p>​ 解决递归调用栈溢出的方法是通过<strong>尾递归</strong>优化。尾递归是指，在函数返回的时候，调用自身本身，并且，return语句不能包含表达式。这样，编译器或者解释器就可以把尾递归做优化，使递归本身无论调用多少次，都只占用一个栈帧，不会出现栈溢出的情况。</p>
<p>例如：</p>
<pre class="language-"><code class="lang-python"><span class="token comment">#尾递归函数的定义</span>
<span class="token keyword">def</span> <span class="token function">fact</span><span class="token punctuation">(</span>n<span class="token punctuation">)</span><span class="token punctuation">:</span>
<span class="token keyword">return</span> fact_iter<span class="token punctuation">(</span>n<span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">)</span>
<span class="token keyword">def</span> <span class="token function">fact_iter</span><span class="token punctuation">(</span>num<span class="token punctuation">,</span> product<span class="token punctuation">)</span><span class="token punctuation">:</span>
<span class="token keyword">if</span> num <span class="token operator">==</span> <span class="token number">1</span><span class="token punctuation">:</span>
<span class="token keyword">return</span> product
<span class="token keyword">return</span> fact_iter<span class="token punctuation">(</span>num <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">,</span> num <span class="token operator">*</span> product<span class="token punctuation">)</span>
<span class="token comment">#计算5的阶乘，递归函数调用细节</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> fact_iter<span class="token punctuation">(</span><span class="token number">5</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> fact_iter<span class="token punctuation">(</span><span class="token number">4</span><span class="token punctuation">,</span> <span class="token number">5</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> fact_iter<span class="token punctuation">(</span><span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">20</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> fact_iter<span class="token punctuation">(</span><span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">60</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> fact_iter<span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">120</span><span class="token punctuation">)</span>
<span class="token operator">==</span><span class="token operator">=</span><span class="token operator">></span> <span class="token number">120</span>
</code></pre>
<p>尾递归调用时，如果做了优化，栈不会增长，无论多少次调用也不会导致栈溢出。但实际上大多数编程语言没有针对尾递归做优化，Python解释器也没有做优化，因此当没有优化的情况下任何递归函数都存在栈溢出的问题。</p>
<p>参考：</p>
<ul>
<li><a href="https://www.liaoxuefeng.com/wiki/1016959663602400/1017268131039072" target="_blank">https://www.liaoxuefeng.com/wiki/1016959663602400/1017268131039072</a> </li>
</ul>
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