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<div class="section" id="recursion">
<h1>Recursion<a class="headerlink" href="#recursion" title="Permalink to this headline"></a></h1>
<p>You’ve seen functions that call other functions.</p>
<p>If a function calls <em>itself</em>, we call that <strong>recursion</strong></p>
<p>Like with other functions, a call within a call establishes a <em>call stack</em></p>
<p>With recursion, if you are not careful, this stack can get <em>very</em> deep.</p>
<p>Python has a maximum limit to how much it can recurse. This is intended to
save your machine from running out of RAM.</p>
<p>Recursion is especially useful for a particular set of problems.</p>
<p>For example, take the case of the <em>factorial</em> function.</p>
<p>In mathematics, the <em>factorial</em> of an integer is the result of multiplying that
integer by every integer smaller than it down to 1.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>5! == 5 * 4 * 3 * 2 * 1
</pre></div>
</div>
<p>We can use a recursive function nicely to model this mathematical function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>1! = 1
2! = 2 * 1 = 2 * 1!
3! = 3 * 2 * 1 = 3 * 2!
</pre></div>
</div>
<p>So we have a pattern here – each value can be defined in terms of the previous value.</p>
<p>So generically:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>1! = 1
n! = n * (n-1)!
</pre></div>
</div>
<p>How would we put that in code? Pretty straightforward translation:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="n">n</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<p>That was pretty easy – what happens when we run it?</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [2]: </span><span class="n">factorial</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">RecursionError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="nn"><ipython-input-2-3fd9b1939623></span> in <span class="ni"><module></span><span class="nt">()</span>
<span class="ne">----> </span><span class="mi">1</span> <span class="n">factorial</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="nn"><ipython-input-1-5ff805d50ea6></span> in <span class="ni">factorial</span><span class="nt">(n)</span>
<span class="g g-Whitespace"> </span><span class="mi">1</span> <span class="k">def</span> <span class="nf">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="ne">----> </span><span class="mi">2</span> <span class="k">return</span> <span class="n">n</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="o">...</span> <span class="n">last</span> <span class="mi">1</span> <span class="n">frames</span> <span class="n">repeated</span><span class="p">,</span> <span class="kn">from</span> <span class="nn">the</span> <span class="n">frame</span> <span class="n">below</span> <span class="o">...</span>
<span class="nn"><ipython-input-1-5ff805d50ea6></span> in <span class="ni">factorial</span><span class="nt">(n)</span>
<span class="g g-Whitespace"> </span><span class="mi">1</span> <span class="k">def</span> <span class="nf">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="ne">----> </span><span class="mi">2</span> <span class="k">return</span> <span class="n">n</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="ne">RecursionError</span>: maximum recursion depth exceeded
</pre></div>
</div>
<p>OOPS! that didn’t work – why not? Let’s add a print…</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"factorial called with"</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="k">return</span> <span class="n">n</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<p>And call it:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [5]: </span><span class="n">factorial</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">factorial called with 3</span>
<span class="go">factorial called with 2</span>
<span class="go">factorial called with 1</span>
<span class="go">factorial called with 0</span>
<span class="go">factorial called with -1</span>
<span class="go">factorial called with -2</span>
<span class="go">factorial called with -3</span>
<span class="go">factorial called with -4</span>
<span class="go">factorial called with -5</span>
<span class="go">...</span>
<span class="go"><ipython-input-3-54c2cb2bf478> in factorial(n)</span>
<span class="go"> 1 def factorial(n):</span>
<span class="go"> 2 print("factorial called with", n)</span>
<span class="go">----> 3 return n * factorial(n-1)</span>
<span class="go">RecursionError: maximum recursion depth exceeded while calling a Python object</span>
</pre></div>
</div>
<p>Now it’s clear what’s going on – each time you call the function, it calls itself with a value one less – but then it just keeps going into the deep negative numbers, and only stops because Python reaches its recursion limit.</p>
<p>This makes clear a core requirement of recursive functions:</p>
<blockquote>
<div><p><strong>Recursive functions must have a termination criteria!</strong></p>
</div></blockquote>
<p>That is, there must be a case (or more than one) for which they return a direct value. What should that be for factorial? Well, it’s part of the definition that 1! == 1 – so let’s put that in our function:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"factorial called with"</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">n</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<p>and try that:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [7]: </span><span class="n">factorial</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">factorial called with 3</span>
<span class="go">factorial called with 2</span>
<span class="go">factorial called with 1</span>
<span class="gh">Out[7]: </span><span class="go">6</span>
</pre></div>
</div>
<p>Much better! Try it out now with various values, and maybe without the print:</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><span class="gp">In [14]: </span><span class="n">factorial</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="gh">Out[14]: </span><span class="go">1</span>
<span class="gp">In [15]: </span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gh">Out[15]: </span><span class="go">2</span>
<span class="gp">In [16]: </span><span class="n">factorial</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="gh">Out[16]: </span><span class="go">6</span>
<span class="gp">In [17]: </span><span class="n">factorial</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gh">Out[17]: </span><span class="go">24</span>
</pre></div>
</div>
<p>Looking good!</p>
<p>Exercise for the reader: What happens if you pass in a negative number?
Think about it first, before you try it. Hint – it won’t work!
How would you change your code to make it more robust?</p>
<div class="section" id="summary">
<h2>Summary<a class="headerlink" href="#summary" title="Permalink to this headline"></a></h2>
<ul class="simple">
<li><p>Whenever you have a function that can be defined in terms of itself, you have a use case for recursion. It can make for nice compact, clear code.</p></li>
<li><p>Python will create a new “stack frame” for each call to the function – so each call is kept separate, with separate local variables.</p></li>
</ul>
<p>But:</p>
<ul class="simple">
<li><p>Python has a limited recursion depth – so it can’t be used for “big” problems.</p></li>
<li><p>You do need to make sure the calls will terminate.</p></li>
</ul>
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