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<h1 id="recursion-on-trees">Recursion on Trees</h1>
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<p>CMPT 225</p>
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<h2 id="recursion">Recursion</h2>
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<p>Recursion: A definition of a function is recursive if the body cointains an application of itself.</p>
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<p>Consider: <span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover><mi>i</mi></mrow><annotation encoding="application/x-tex">
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S(n)=\sum_{i=0}^{n} i
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span></span></span></span></span></p>
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<p>or</p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>n</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>n</mi><mo>+</mo><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if </mtext><mi>n</mi><mo>></mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">
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S(n) = \begin{cases}
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0 & \text{if } n=0\\
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n+S(n-1) & \text{if } n>0
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\end{cases}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if </span></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
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<p>These two descriptions of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> suggest two implementations:</p>
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<p>e.g.:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>S(n){
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s=0
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for i=1..n
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s=s+i
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return s
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}
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</code></pre></div></div>
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<p>or</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>S(n){
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if n=0
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return 0
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else
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return n+S(n-1)
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}
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</code></pre></div></div>
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<h2 id="recursive-version">Recursive Version</h2>
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<ul>
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<li>-> S(4)
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<ul>
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<li>-> S(3)
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<ul>
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<li>-> S(2)
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<ul>
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<li>-> S(1)
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<ul>
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<li>-> S(0)</li>
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<li><span id="inner0">returns 0</span></li>
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</ul>
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</li>
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<li>returns <span id="inner1">1+<a href="#inner0">0</a></span></li>
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</ul>
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</li>
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<li>returns <span id="inner2">2+<a href="#inner1">1</a>=3</span></li>
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</ul>
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</li>
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<li>returns <span id="inner3">3+<a href="#inner2">3</a>=6</span></li>
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</ul>
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</li>
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<li>returns 4+<a href="#inner3">6</a>=10</li>
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</ul>
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<p>Iterative version:</p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mo>=</mo><mn>0</mn><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>i</mi><mo>=</mo><mn>1</mn><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover><mi>i</mi><mo>=</mo><mn>3</mn><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>3</mn></mrow><mi>n</mi></munderover><mi>i</mi><mo>=</mo><mo>…</mo><mspace linebreak="newline"></mspace><mo>=</mo><mn>0</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">
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S = 0 + \sum_{i=1}^{n} i = 1 + \sum_{i=2}^{n} i = 3 + \sum_{i=3}^{n} i = \dots\\
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= 0+1+2+3+4
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">3</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.12em;vertical-align:0em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span></p>
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<p>The <em>same computation</em>, but a different control strategy.</p>
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<h2 id="recursion--the-call-stack">Recursion & The Call Stack</h2>
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<p>Code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>S(n){
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if(n=0){
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r=0
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}else{
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r=n+S(n-1)
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}
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return r
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}
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</code></pre></div></div>
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<p>Compute <code class="language-plaintext highlighter-rouge">p=S(2)</code>:</p>
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<ul>
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<li>Call S(2):
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<ul>
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<li>n = 2</li>
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<li>r = ? ->
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<ul>
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<li>Call S(1):
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<ul>
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<li>n=1</li>
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<li>r=? ->
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<ul>
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<li>Call S(0):
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<ul>
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<li>N=0</li>
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<li>r=0</li>
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<li><span id="innerstack1">return 0</span></li>
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</ul>
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</li>
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</ul>
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</li>
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<li><span id="innerstack2">return 1+<a href="#innerstack1">S(0)</a></span></li>
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</ul>
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</li>
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</ul>
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</li>
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<li>return 2+<a href="#innerstack2">S(1)</a></li>
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</ul>
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</li>
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</ul>
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<p>The call stack at the end:</p>
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<table>
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<tbody>
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<tr>
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<td>Name</td>
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<td>Value</td>
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<td>Removed</td>
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</tr>
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<tr>
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<td>S</td>
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<td><code of S; n=0, r=0></td>
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<td>true</td>
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</tr>
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<tr>
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<td>S</td>
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<td><code of S; n=1, r=1></td>
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<td>true</td>
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</tr>
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<tr>
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<td>S</td>
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<td><code of S; n=2, r=3></td>
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<td>true</td>
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|
|
</tr>
|
|
|
<tr>
|
|
|
<td>p</td>
|
|
|
<td>3</td>
|
|
|
<td>false</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>After the call the <code class="language-plaintext highlighter-rouge">S(2)</code> is complete, the entire call stack of S to S to S is gone.</p>
|
|
|
|
|
|
<p>There are 2 more slides containing <em>slightly</em> different versions of the same diagrams.
|
|
|
They do not provide any extra information.</p>
|
|
|
|
|
|
<h2 id="recursion-on-trees-1">Recursion on Trees</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>We will often use recursion & induction on trees.
|
|
|
<ul>
|
|
|
<li>e.g. the tree rooted a v has some property if its subtrees have some related property</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>e.g. the height of node v in a binary tree may be defined by:</li>
|
|
|
</ul>
|
|
|
|
|
|
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> if v is a leaf</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>+</mo><mtext>max</mtext><mo stretchy="false">{</mo><mi>h</mi><mo stretchy="false">(</mo><mtext>left</mtext><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>h</mi><mo stretchy="false">(</mo><mtext>right</mtext><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">
|
|
|
h(v) = \begin{cases}
|
|
|
0 & \text{ if v is a leaf}\\
|
|
|
1 + \text{max}\{h(\text{left}(v)), h(\text{right}(v))\} & \text{ otherwise}
|
|
|
\end{cases}
|
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">{</span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord text"><span class="mord">left</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord text"><span class="mord">right</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mclose">)</span><span class="mclose">}</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> if v is a leaf</span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> otherwise</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
|
|
|
|
|
|
<p>(We can define h(left(v)) to be -1 if left(v) does not exist, and sim. for right(v)).</p>
|
|
|
|
|
|
<h2 id="recurssion-a-trees-examples">Recurssion a Trees Examples</h2>
|
|
|
|
|
|
<p>height of node v in T:</p>
|
|
|
|
|
|
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> if v is a leaf</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>+</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">{</mo><mi>h</mi><mo stretchy="false">(</mo><mtext>left</mtext><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>h</mi><mo stretchy="false">(</mo><mtext>right</mtext><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> ow</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">
|
|
|
h(v) = \begin{cases}
|
|
|
0 & \text{ if v is a leaf}\\
|
|
|
1+ max\{h(\text{left}(v)),h(\text{right}(v))\} & \text{ ow}
|
|
|
\end{cases}
|
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">m</span><span class="mord mathdefault">a</span><span class="mord mathdefault">x</span><span class="mopen">{</span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord text"><span class="mord">left</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">h</span><span class="mopen">(</span><span class="mord text"><span class="mord">right</span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mclose">)</span><span class="mclose">}</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> if v is a leaf</span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord"> ow</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
|
|
|
|
|
|
<p>for the follwing tree: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">h(v)=?</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
v
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
a
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
e
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
g
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
h
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
f
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
i
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
b
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
c
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
d
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<h2 id="recursion-on-trees-example-pt-2">Recursion on Trees Example (pt 2)</h2>
|
|
|
|
|
|
<p>(See math equation of previous slide)</p>
|
|
|
|
|
|
<p>h(v) = 3</p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
v (3)
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
a (2)
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
e (1)
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
g (0)
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
h (0)
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
f (1)
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
i (0)
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
b (1)
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
c (0)
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
d (0)
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<h2 id="pseudo-code-version">Pseudo-code version:</h2>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>height(v){
|
|
|
if v is a leaf
|
|
|
return 0
|
|
|
if v has one child u
|
|
|
return 1+height(u)
|
|
|
else
|
|
|
return 1+max(height(left(v)), height(right(v)))
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<h2 id="traversals-of-binary-tree">Traversals of Binary Tree</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>A traversal of a graph is a process that “visits” each node in the graph once.</li>
|
|
|
<li>We consider 4 standard tree traversalt:
|
|
|
<ol>
|
|
|
<li>level order</li>
|
|
|
<li>pre-order</li>
|
|
|
<li>in-order</li>
|
|
|
<li>post-order</li>
|
|
|
</ol>
|
|
|
</li>
|
|
|
<li>2,3,4 begin at the root & recursively visit the nodes in each subtree & the root. They vary in the relative ???(can’t read).</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>(Level order later)</p>
|
|
|
|
|
|
<p>Code:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>pre-order-T(v){
|
|
|
visit v
|
|
|
pre-order-T(left(v))
|
|
|
pre-order-T(right(v))
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>pre-order-T(v) does nothing if v does not exist.</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>v is visited before any of its decendants</li>
|
|
|
<li>every node in the left subtree is visited before any node in the right subtree.</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Tree to come back to:</p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
A
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
B
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
D
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
E
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
C
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span>
|
|
|
F
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<p>Pre-order-traversal: A,B,D,E,C,F</p>
|
|
|
|
|
|
<h2 id="in-order-t">in-order-T</h2>
|
|
|
|
|
|
<p>code:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>in-order-T(v){
|
|
|
in-order-T(left(v))
|
|
|
visit v
|
|
|
in-order-T(right(v))
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>In order traversal: D,B,E,A,C,F</p>
|
|
|
|
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<h2 id="post-order-t">post-order-T</h2>
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<p>code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>post-order-T(v){
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post-order-T(left(v))
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post-order-T(right(v))
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visit v
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}
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</code></pre></div></div>
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<p>Post order traversal: D,E,B,F,C,A</p>
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<h2 id="end">End</h2>
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<p>…some repeated slides… unknown reason</p>
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