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<h1 id="rooted-trees--cmpt-225">Rooted Trees – CMPT 225</h1>
<h2 id="graphs">Graphs</h2>
<p>Graph is a pair G=&lt;V,E&gt;, with:</p>
<ul>
<li>V, a set called “vertacies” or “nodes”</li>
<li>E, a set of pairs from V, i.e. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>⊆</mo><mi>V</mi><mo>×</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">E \subseteq V\times V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">E</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:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</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:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span> called edges</li>
<li>Example: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">G=\langle\{1,2,3\},\{(1,2),(1,3)\}\rangle</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">G</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:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">⟩</span></span></span></span></li>
</ul>
<p>Example in table format:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g1-1">1</span></td>
<td><a href="#g1-2">2</a>,<a href="#g1-3">3</a></td>
</tr>
<tr>
<td><span id="g1-2">2</span></td>
<td><a href="#g1-1">1</a></td>
</tr>
<tr>
<td><span id="g1-3">3</span></td>
<td><a href="#g1-1">1</a></td>
</tr>
</tbody>
</table>
<p>“G is directed”: edges are ordered pairs (often called arcs)</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo><mo mathvariant="normal">≠</mo><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">
\langle\{\{1,2,3\},\{(1,2),(1,3)\}\}\rangle
\neq
\langle\{\{1,2,3\},\{(2,1),(1,3)\}\}\rangle
</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="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</span><span class="mclose">⟩</span></span></span></span></span></p>
<p>i.e. graph one:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g2-1">1</span></td>
<td><a href="#g2-2">2</a>,<a href="#g2-3">3</a></td>
</tr>
<tr>
<td><span id="g2-2">2</span></td>
<td> </td>
</tr>
<tr>
<td><span id="g2-3">3</span></td>
<td> </td>
</tr>
</tbody>
</table>
<p>does not equal graph two:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g3-1">1</span></td>
<td><a href="#g3-3">3</a></td>
</tr>
<tr>
<td><span id="g3-2">2</span></td>
<td><a href="#g3-1">1</a></td>
</tr>
<tr>
<td><span id="g3-3">3</span></td>
<td> </td>
</tr>
</tbody>
</table>
<p>“G is undirected” edges are sets</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">⟨</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo stretchy="false">}</mo><mo separator="true">,</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">
\langle\{\{1,2,3\},\{(2,1),(1,3)\}\}\rangle
=
\langle\{\{1,2,3\},\{(1,2),(3,1)\}\}\rangle
</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="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</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:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mopen">{</span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">{</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mclose">}</span><span class="mclose">}</span><span class="mclose">⟩</span></span></span></span></span></p>
<p>i.e. graph one:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g4-1">1</span></td>
<td><a href="#g4-3">3</a>,<a href="#g4-2">2</a></td>
</tr>
<tr>
<td><span id="g4-2">2</span></td>
<td><a href="#g4-1">1</a></td>
</tr>
<tr>
<td><span id="g4-3">3</span></td>
<td><a href="#g4-1">1</a></td>
</tr>
</tbody>
</table>
<p>is equal to graph two:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g5-1">1</span></td>
<td><a href="#g5-2">2</a>,<a href="#g5-3">3</a></td>
</tr>
<tr>
<td><span id="g5-2">2</span></td>
<td><a href="#g5-1">1</a></td>
</tr>
<tr>
<td><span id="g5-3">3</span></td>
<td><a href="#g5-1">1</a></td>
</tr>
</tbody>
</table>
<p>By default, by “graph” we will mean “undirected graph”.</p>
<h2 id="path">Path</h2>
<p>Path of G of length n is a sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>…</mo><msub><mi>V</mi><mi>n</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle V_{0},V_{1},V_{2},\dots V_{n}\rangle</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="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> of vertacies s-t. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>V</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><msub><
<p>s-t Path in G: path <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,\dots,t\rangle</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="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">t</span><span class="mclose">⟩</span></span></span></span> in G.</p>
<p>Example of a s-t path of length 6 for G: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>e</mi><mo separator="true">,</mo><mi>f</mi><mo separator="true">,</mo><mi>h</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,d,c,e,f,h,t\rangle</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="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">t</span><span class="mclose">⟩</span></span></span></span></p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g6-a">a</span></td>
<td><a href="#g6-b">b</a>,<a href="#g6-s">s</a></td>
</tr>
<tr>
<td><span id="g6-b">b</span></td>
<td><a href="#g6-a">a</a>,<a href="#g6-c">c</a></td>
</tr>
<tr>
<td><span id="g6-c">c</span></td>
<td><a href="#g6-b">b</a>,<a href="#g6-d">d</a>,<a href="#g6-e">e</a>,<a href="#g6-s">s</a></td>
</tr>
<tr>
<td><span id="g6-d">d</span></td>
<td><a href="#g6-c">c</a>,<a href="#g6-s">s</a></td>
</tr>
<tr>
<td><span id="g6-e">e</span></td>
<td><a href="#g6-c">c</a>,<a href="#g6-f">f</a>,<a href="#g6-g">g</a></td>
</tr>
<tr>
<td><span id="g6-f">f</span></td>
<td><a href="#g6-e">e</a>,<a href="#g6-h">h</a></td>
</tr>
<tr>
<td><span id="g6-g">g</span></td>
<td><a href="#g6-e">e</a>,<a href="#g6-h">h</a>,<a href="#g6-t">t</a></td>
</tr>
<tr>
<td><span id="g6-h">h</span></td>
<td><a href="#g6-f">f</a>,<a href="#g6-g">g</a>,<a href="#g6-t">t</a></td>
</tr>
<tr>
<td><span id="g6-s">s</span></td>
<td><a href="#g6-a">a</a>,<a href="#g6-c">c</a>,<a href="#g6-d">d</a></td>
</tr>
<tr>
<td><span id="g6-t">t</span></td>
<td><a href="#g6-g">g</a>,<a href="#g6-h">h</a></td>
</tr>
</tbody>
</table>
<p>Vertex t is <em>reachable</em> from s in G if ther is an s-t path in G.
G is <em>connected</em> if for every pair <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">u,v\in V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</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:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>, u is reachable from v.</p>
<p>Transcriber’s note: this is much easier to show with graphics and in fact very hard to show easily without graphics. I did my best, but it should be noted that this is <em>very</em> obvious with a visual graph.
It becomes two seperate graphs, in essense.
So for convenience, I will be splitting this up into two tables, even though technically it’s one “graph” with two sets of connected nodes.</p>
<p id="connected1">Connected:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g7-a">a</span></td>
<td><a href="#g7-b">b</a>,<a href="#g7-s">s</a></td>
</tr>
<tr>
<td><span id="g7-b">b</span></td>
<td><a href="#g7-a">a</a>,<a href="#g7-c">c</a></td>
</tr>
<tr>
<td><span id="g7-c">c</span></td>
<td><a href="#g7-b">b</a>,<a href="#g7-d">d</a>,<a href="#g7-e">e</a>,<a href="#g7-s">s</a></td>
</tr>
<tr>
<td><span id="g7-d">d</span></td>
<td><a href="#g7-c">c</a>,<a href="#g7-s">s</a></td>
</tr>
<tr>
<td><span id="g7-e">e</span></td>
<td><a href="#g7-c">c</a>,<a href="#g7-f">f</a>,<a href="#g7-g">g</a></td>
</tr>
<tr>
<td><span id="g7-f">f</span></td>
<td><a href="#g7-e">e</a>,<a href="#g7-h">h</a></td>
</tr>
<tr>
<td><span id="g7-g">g</span></td>
<td><a href="#g7-e">e</a>,<a href="#g7-h">h</a>,<a href="#g7-t">t</a></td>
</tr>
<tr>
<td><span id="g7-h">h</span></td>
<td><a href="#g7-f">f</a>,<a href="#g7-g">g</a>,<a href="#g7-t">t</a></td>
</tr>
<tr>
<td><span id="g7-s">s</span></td>
<td><a href="#g7-a">a</a>,<a href="#g7-c">c</a>,<a href="#g7-d">d</a></td>
</tr>
<tr>
<td><span id="g7-t">t</span></td>
<td><a href="#g7-g">g</a>,<a href="#g7-h">h</a></td>
</tr>
</tbody>
</table>
<p>Not connected:</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g9-a">a</span></td>
<td><a href="#g9-b">b</a>,<a href="#g9-s">s</a></td>
</tr>
<tr>
<td><span id="g9-b">b</span></td>
<td><a href="#g9-a">a</a>,<a href="#g9-c">c</a></td>
</tr>
<tr>
<td><span id="g9-c">c</span></td>
<td><a href="#g9-b">b</a>,<a href="#g9-d">d</a>,<a href="#g9-e">e</a>,<a href="#g9-s">s</a></td>
</tr>
<tr>
<td><span id="g9-d">d</span></td>
<td><a href="#g9-c">c</a>,<a href="#g9-s">s</a></td>
</tr>
<tr>
<td><span id="g9-e">e</span></td>
<td><a href="#g9-c">c</a></td>
</tr>
<tr>
<td><span id="g9-s">s</span></td>
<td><a href="#g9-a">a</a>,<a href="#g9-c">c</a>,<a href="#g9-d">d</a></td>
</tr>
</tbody>
</table>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g10-f">f</span></td>
<td><a href="#g10-h">h</a></td>
</tr>
<tr>
<td><span id="g10-g">g</span></td>
<td><a href="#g10-h">h</a>,<a href="#g10-t">t</a></td>
</tr>
<tr>
<td><span id="g10-h">h</span></td>
<td><a href="#g10-f">f</a>,<a href="#g10-g">g</a>,<a href="#g10-t">t</a></td>
</tr>
<tr>
<td><span id="g10-t">t</span></td>
<td><a href="#g10-g">g</a>,<a href="#g10-h">h</a></td>
</tr>
</tbody>
</table>
<h2 id="cycles--trees">Cycles &amp; Trees</h2>
<p>Cycle in G: path <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><mo>…</mo><msub><mi>V</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>V</mi><mi>n</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle V_{0},\dots V_{n-1},V_{n}\rangle</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="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> in G where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mn>0</mn></msub><mo>=</mo><msub><mi>V</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">V_{0}=V_{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></spa
<p><em>Simple path</em>: all vertecies are distinct.
<em>Simple Cycle</em>: cycle <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><mo>…</mo><msub><mi>V</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>V</mi><mi>n</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle V_{0},\dots V_{n-1},V_{n}\rangle</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="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">⟩</span></span></span></span> where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>V</mi><mn>0</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>V</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\langle V_{0},\dots,V_{n-1}</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="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0
<p>Simple cylcle of length 5: (see <a href="#connected1">“connected” graph</a> for confirmation): <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>a</mi><mo separator="true">,</mo><mi>s</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,d,c,b,a,s\rangle</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="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">s</span><span class="mclose">⟩</span></span></span></span></p>
<p>Cycle of length 7 + repeats: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>f</mi><mo separator="true">,</mo><mi>e</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>s</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,d,c,b,f,e,c,s\rangle</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="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">s</span><span class="mclose">⟩</span></span></span></span></p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g11-a">a</span></td>
<td><a href="#g11-b">b</a>,<a href="#g11-s">s</a></td>
</tr>
<tr>
<td><span id="g11-b">b</span></td>
<td><a href="#g11-a">a</a>,<a href="#g11-c">c</a>,<a href="#g11-f">f</a></td>
</tr>
<tr>
<td><span id="g11-c">c</span></td>
<td><a href="#g11-b">b</a>,<a href="#g11-d">d</a>,<a href="#g11-e">e</a>,<a href="#g11-f">f</a>,<a href="#g11-s">s</a></td>
</tr>
<tr>
<td><span id="g11-d">d</span></td>
<td><a href="#g11-c">c</a>,<a href="#g11-s">s</a></td>
</tr>
<tr>
<td><span id="g11-e">e</span></td>
<td><a href="#g11-c">c</a>,<a href="#g11-f">f</a>,<a href="#g11-g">g</a></td>
</tr>
<tr>
<td><span id="g11-f">f</span></td>
<td><a href="#g11-c">c</a>,<a href="#g11-e">e</a>,<a href="#g11-h">h</a>,<a href="#g11-b">b</a></td>
</tr>
<tr>
<td><span id="g11-g">g</span></td>
<td><a href="#g11-e">e</a>,<a href="#g11-h">h</a>,<a href="#g11-t">t</a></td>
</tr>
<tr>
<td><span id="g11-h">h</span></td>
<td><a href="#g11-f">f</a>,<a href="#g11-g">g</a>,<a href="#g11-t">t</a></td>
</tr>
<tr>
<td><span id="g11-s">s</span></td>
<td><a href="#g11-a">a</a>,<a href="#g11-c">c</a>,<a href="#g11-d">d</a></td>
</tr>
<tr>
<td><span id="g11-t">t</span></td>
<td><a href="#g11-g">g</a>,<a href="#g11-h">h</a></td>
</tr>
</tbody>
</table>
<h2 id="tree">Tree</h2>
<p>Tree: a connected, acyclic graph:</p>
<p>Example 1 (not a tree, due to cycle: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⟨</mo><mi>s</mi><mo separator="true">,</mo><mi>c</mi><mo separator="true">,</mo><mi>d</mi><mo separator="true">,</mo><mi>s</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle s,c,d,s\rangle</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="mopen">⟨</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">s</span><span class="mclose">⟩</span></span></span></span>):</p>
<table>
<thead>
<tr>
<th>node</th>
<th>connections</th>
</tr>
</thead>
<tbody>
<tr>
<td><span id="g12-a">a</span></td>
<td><a href="#g12-b">b</a></td>
</tr>
<tr>
<td><span id="g12-b">b</span></td>
<td><a href="#g12-a">a</a>,<a href="#g12-c">c</a></td>
</tr>
<tr>
<td><span id="g12-c">c</span></td>
<td><a href="#g12-b">b</a>,<a href="#g12-d">d</a>,<a href="#g12-e">e</a>,<a href="#g12-s">s</a></td>
</tr>
<tr>
<td><span id="g12-d">d</span></td>
<td><a href="#g12-c">c</a>,<a href="#g12-s">s</a></td>
</tr>
<tr>
<td><span id="g12-e">e</span></td>
<td><a href="#g12-c">c</a>,<a href="#g12-g">g</a></td>
</tr>
<tr>
<td><span id="g12-f">f</span></td>
<td><a href="#g12-h">h</a></td>
</tr>
<tr>
<td><span id="g12-g">g</span></td>
<td><a href="#g12-e">e</a>,<a href="#g12-h">h</a>,<a href="#g12-t">t</a></td>
</tr>
<tr>
<td><span id="g12-h">h</span></td>
<td><a href="#g12-f">f</a>,<a href="#g12-g">g</a>,<a href="#g12-t">t</a></td>
</tr>
<tr>
<td><span id="g12-s">s</span></td>
<td><a href="#g12-c">c</a>,<a href="#g12-d">d</a></td>
</tr>
<tr>
<td><span id="g12-t">t</span></td>
<td><a href="#g12-g">g</a>,<a href="#g12-h">h</a></td>
</tr>
</tbody>
</table>
<p>Example 2 (a tree, no cycle, all connected):</p>
<p>Now using ARIA trees:</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
s
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
a
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
d
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
c
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
e
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
h
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
g
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
i
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Example 3 (not a tree) (transcriber’s note: written as two trees for simplicity’s sake)</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
s
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
a
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
d
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
c
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
e
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
h
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
g
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
i
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Fact: every tree with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span></span></span></span> verticies has <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">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><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> edges.</p>
<h2 id="rooted-tree">Rooted-tree:</h2>
<p>Rooted tree: tree with a distringuished vertex called the root.</p>
<p>Unrooted tree:</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>
c
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
d
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
e
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
g
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
h
</span>
</li>
</ul>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
i
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
j
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
k
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
l
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Tree T with root:</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
a (root)
</span>
<ul role="group">
<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>
<li role="treeitem" tabindex="-1">
<span>
e
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
f
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
g
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
h
</span>
</li>
</ul>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
i
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
j
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
k
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
l
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>An alternate drawing of T, with no semantic change.</p>
<p>Notice: in a tree there is a unique path between any two verticies.
So: a unique path from any vertex to the root.
Thus: the root indicates a <em>direction</em> on the edges e.g. towards the root.</p>
<p>A graph of the same tree, T, with arrows pointing towards the root.
There is no reason for having any semantic markup for this.</p>
<p>(sometimes “away from the root”).</p>
<h2 id="rooted-tree-terminology">Rooted Tree Terminology</h2>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
z (root, anscestor of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
y
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
x (leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
w (leaf)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
v (anscestor of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
u (leaf)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
t (anscestor of a, parent of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
s (sibling of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
r (sibling of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
a (descendant of a, anscestor of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
q (child of a, desandant of a)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
o (decendant of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
n (decendant of a, leaf)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
m (decendant of a, leaf)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
p (child of a, desendant of a, leaf)
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<ul>
<li>The <em>root</em> has no parent.</li>
<li><em>Leaves</em> have no children.</li>
<li>Internal nodes are the non-leaves (sometimes root ??? [can’t read] too)</li>
</ul>
<h2 id="depth--height">Depth &amp; Height</h2>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
4 (depth 0)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1 (depth 1)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
0 (depth 2)
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
0 (depth 2)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
3 (depth 1)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
2 (depth 2)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1 (depth 3)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
0 (depth 4)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
0 (depth 3)
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
0 (depth 2)
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<ul>
<li>Depth of node v = length of path from v to the root.</li>
<li>Height of node v = length of longest path from to a decendant of v (e.g. to a leaf)</li>
<li>Height of Tree T = height of its root = max height of any node in T = max depth of any node in T.</li>
</ul>
<p>A rooted tree is:</p>
<ul>
<li>k-ary if <em>no node has</em> &gt;k children.</li>
<li>binary if <em>no node has</em> &gt;2 children.</li>
<li>ordered if the children of every node are ordered.</li>
</ul>
<p>E.g. A ordered trenary tree:</p>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
no name
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
2
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
2
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
3
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
1
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
2
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
3
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<p>Notice: when we draw a tree, or represent it in a data structure, we order it.</p>
<p>In an ordered binary tree, every child of a node v is either the “left child of v” or the “right child of v”.</p>
<p>Transcriber’s note: nodes’ children are transcribes from left to right. So for a binary tree, the first child is the “left child” and the second child is the “right child”.</p>
<h2 id="subtree-rooted-at-v">Subtree rooted at v</h2>
<p>Subtree rooted at v: tree with root v and containing all decendants of v.</p>
<p>In a binary tree:</p>
<ul>
<li>“left subtree of v” means the subtree rooted at the left child of v.</li>
<li>sim. for “right child of v”.</li>
</ul>
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
<ul role="tree">
<li role="treeitem" tabindex="-1">
<span>
z
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
v (subtree rooted at v)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
y (subtree rooted at v, left subtree of v)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
v (subtree rooted at v, left subtree of v
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
u (subtree rooted at v, left subtree of v
</span>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
x (subtree rooted at v, right subtree of v)
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
w (subtree rooted at v, right subtree of v)
</span>
</li>
</ul>
</li>
</ul>
</li>
<li role="treeitem" tabindex="-1">
<span>
a
</span>
<ul role="group">
<li role="treeitem" tabindex="-1">
<span>
b
</span>
</li>
<li role="treeitem" tabindex="-1">
<span>
c
</span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<!-- END OF GENERATED CODE -->
<h2 id="end">End</h2>
</div>
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