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<title data-rh="true">On the rules of the red-black tree | mf</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://blog.mfocko.xyz/algorithms/rb-trees/rules/"><meta data-rh="true" property="og:locale" content="en"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docusaurus_version" content="current"><meta data-rh="true" name="docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" name="docsearch:version" content="current"><meta data-rh="true" name="docsearch:docusaurus_tag" content="docs-algorithms-current"><meta data-rh="true" property="og:title" content="On the rules of the red-black tree | mf"><meta data-rh="true" name="description" content="Shower thoughts on the rules of the red-black tree.
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<p>Have you ever thought about the red-black tree rules in more depth? Why are they
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formulated the way they are? How come they keep the tree balanced? Let's go through
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each of the red-black tree rules and try to change, break and contemplate about
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them.</p>
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<p>We expect that you are familiar with the following set of the rules<sup><a href="#user-content-fn-1" id="user-content-fnref-1" data-footnote-ref="true" aria-describedby="footnote-label">1</a></sup>:</p>
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<ol>
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<li>Every node is either red or black.</li>
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<li>The root is black.</li>
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<li>Every leaf (<code>nil</code>) is black.</li>
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<li>If a node is red, then both its children are black.</li>
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<li>For each node, all simple paths from the node to descendant leaves contain the
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same number of black nodes.</li>
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</ol>
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<p>Each section will go into <em>reasonable</em> details of each rule.</p>
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="1ª-every-node-is-either-red-or-black">1ª Every node is either red or black.<a href="#1ª-every-node-is-either-red-or-black" class="hash-link" aria-label="Direct link to 1ª Every node is either red or black." title="Direct link to 1ª Every node is either red or black."></a></h2>
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<p>OK… This one is very simple. It is just a definition and is used in all other
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rules. Not much to talk about here. Or is there?</p>
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<h3 class="anchor anchorWithStickyNavbar_LWe7" id="do-i-really-need-the-nodes-to-be-explicitly-colored">Do I really need the nodes to be explicitly colored?<a href="#do-i-really-need-the-nodes-to-be-explicitly-colored" class="hash-link" aria-label="Direct link to Do I really need the nodes to be explicitly colored?" title="Direct link to Do I really need the nodes to be explicitly colored?"></a></h3>
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<p>The answer is no. Balancing of the red-black trees is “enforced” by the 4th and
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5th rule in the enumeration above. There are many ways you can avoid using colors.</p>
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<h4 class="anchor anchorWithStickyNavbar_LWe7" id="black-height">Black height<a href="#black-height" class="hash-link" aria-label="Direct link to Black height" title="Direct link to Black height"></a></h4>
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<p>We mentioned the 4th and 5th rule and that it enforces the balancing. What does
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it mean for us?</p>
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<p>Well, we definitely do not have to use the colors, which even as a <em>boolean</em> flag
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would take at least 1 byte of space (and usually even more), cause… well, it is
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easier for the CPU to work with words rather than single bits.</p>
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<p>We could use the black height, couldn't we? It would mean more memory used, cause
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it should be ideally big and unsigned. Can we tell the color of a node from the
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black height? Of course we can, if my child has the same black height as I do,
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it means that there was no black node added on the path between us and therefore
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my child would be colored red.</p>
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<p>Example of a red-black tree that keeps count of black nodes on paths to the
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leaves follows:</p>
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<p><img loading="lazy" alt="Red-black tree with black height" src="/assets/images/rb_height_light-0aff6e7a40a9f601e0dd1114e43e43b1.svg#gh-light-mode-only" width="923" height="539" class="img_ev3q">
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<img loading="lazy" alt="Red-black tree with black height" src="/assets/images/rb_height_dark-921b2d98d9fe1e579474faf36486f281.svg#gh-dark-mode-only" width="923" height="539" class="img_ev3q"></p>
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<p>We mark the <em>black heights</em> in superscript. You can see that all leaves have the
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black height equal to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span>. Let's take a look at some of the interesting cases:</p>
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<ul>
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<li>
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<p>If we take a look at the node with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>key</mtext><mo>=</mo><mn>9</mn></mrow><annotation encoding="application/x-tex">\text{key} = 9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord text"><span class="mord">key</span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">9</span></span></span></span>, we can see that it is
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coloured red and its black height is 1, because it is a leaf.</p>
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<p>Let's look at its parent (node with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>key</mtext><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">\text{key} = 8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord text"><span class="mord">key</span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">8</span></span></span></span>). On its left side it has
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<code>nil</code> and on its right side the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn></mrow><annotation encoding="application/x-tex">9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">9</span></span></span></span>. And its black height is still <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span>, cause
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except for the <code>nil</code> leaves, there are no other black nodes.</p>
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<p>We can clearly see that if a node has the same black height as its parent, it
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is a red node.</p>
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</li>
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<li>
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<p>Now let's take a look at the root with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>key</mtext><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\text{key} = 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord text"><span class="mord">key</span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">3</span></span></span></span>. It has a black height
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of 3. Both of its children are black nodes and have black height of 2.</p>
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<p>We can see that if a node has its height 1 lower than its parent, it is a black
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node.</p>
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<p>The reasoning behind it is rather simple, we count the black nodes all the way
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to the leaves, therefore if my parent has a higher black height, it means that
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on the path from me to my parent there is a black node, but the only node added
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is me, therefore I must be black.</p>
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</li>
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</ul>
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<h4 class="anchor anchorWithStickyNavbar_LWe7" id="isomorphic-trees">Isomorphic trees<a href="#isomorphic-trees" class="hash-link" aria-label="Direct link to Isomorphic trees" title="Direct link to Isomorphic trees"></a></h4>
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<p>One of the other ways to avoid using color is storing the red-black tree in some
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isomorphic tree. The structure of 2-3-4 tree allows us to avoid using the color
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completely. This is a bit different approach, cause we would be basically using
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different tree, so we keep this note in just as a “hack”.</p>
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="2ª-the-root-is-black">2ª The root is black.<a href="#2ª-the-root-is-black" class="hash-link" aria-label="Direct link to 2ª The root is black." title="Direct link to 2ª The root is black."></a></h2>
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<p>This rule might seem like a very important one, but overall is not. You can safely
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omit this rule, but you also need to deal with the consequences.</p>
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<p>Let's refresh our memory with the algorithm of <em>insert fixup</em>:</p>
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<div class="codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#000000;--prism-background-color:#ffffff"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-text codeBlock_bY9V thin-scrollbar" style="color:#000000;background-color:#ffffff"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#000000"><span class="token plain">WHILE z.p.color == Red</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> IF z.p == z.p.p.left</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> y = z.p.p.right</span><br></span><span class="token-line" style="color:#000000"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> IF y.color == Red</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> z.p.color = Black</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> y.color = Black</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> z.p.p.color = Red</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> z = z.p.p</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> ELSE</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> IF z == z.p.right</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> z = z.p</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> Left-Rotate(T, z)</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> z.p.color = Black</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> z.p.p.color = Red</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> Right-Rotate(T, z.p.p)</span><br></span><span class="token-line" style="color:#000000"><span class="token plain"> ELSE (same as above with “right” and “left” exchanged)</span><br></span><span class="token-line" style="color:#000000"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#000000"><span class="token plain">T.root.color = Black</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
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<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_BuS1"><p>If you have tried to implement any of the more complex data structures, such as
|
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|
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red-black trees, etc., in a statically typed language that also checks you for
|
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|
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<code>NULL</code>-correctness (e.g. <em>mypy</em> or even C# with nullable reference types), you
|
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|
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might have run into numerous issues in the cases where you are 100% sure that you
|
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|
|
cannot obtain <code>NULL</code> because of the invariants, but the static type checking
|
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|
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doesn't know that.</p><p>The issue we hit with the <em>insert fixup</em> is very similar.</p></div></div>
|
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|
|
<p>You might not realize the issue at the first sight, but the algorithm described
|
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|
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with the pseudocode above expects that the root of the red-black tree is black by
|
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|
|
both relying on the invariant in the algorithm and afterwards by enforcing the
|
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|
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black root property.</p>
|
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|
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<p>If we decide to omit this condition, we need to address it in the pseudocodes
|
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accordingly.</p>
|
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|
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<table><thead><tr><th style="text-align:center">Usual algorithm with black root</th><th style="text-align:center">Allowing red root</th></tr></thead><tbody><tr><td style="text-align:center"><img loading="lazy" alt="1ª insertion" src="data:image/svg+xml;base64,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#gh-light-mode-only" width="179" height="155" class="img_ev3q"><img loading="lazy" alt="1ª insertion" src="data:image/svg+xml;base64,PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+CjwhRE9DVFlQRSBzdmcgUFVCTElDICItLy9XM0MvL0RURCBTVkcgMS4xLy9FTiIKICJodHRwOi8vd3d3LnczLm9yZy9HcmFwaGljcy9TVkcvMS4xL0RURC9zdmcxMS5kdGQiPgo8IS0tIEdlbmVyYXRlZCBieSBncmFwaHZpeiB2ZXJzaW9uIDIuNDMuMCAoMCkKIC0tPgo8IS0tIFRpdGxlOiBSQlRyZWUgUGFnZXM6IDEgLS0+Cjxzdmcgd2lkdGg9IjEzNHB0IiBoZWlnaHQ9IjExNnB0Igogdmlld0JveD0iMC4wMCAwLjAwIDEzNC4wMCAxMTYuMDAiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgeG1sbnM6eGxpbms9Imh0dHA6Ly93d3cudzMub3JnLzE5OTkveGxpbmsiPgo8ZyBpZD0iZ3JhcGgwIiBjbGFzcz0iZ3JhcGgiIHRyYW5zZm9ybT0ic2NhbGUoMSAxKSByb3RhdGUoMCkgdHJhbnNsYXRlKDQgMTEyKSI+Cjx0aXRsZT5SQlRyZWU8L3RpdGxlPgo8cG9seWdvbiBmaWxsPSIjMWIxYjFkIiBzdHJva2U9InRyYW5zcGFyZW50IiBwb2ludHM9Ii00LDQgLTQsLTExMiAxMzAsLTExMiAxMzAsNCAtNCw0Ii8+CjwhLS0gMTQwMzk5ODA1NjAzMjgwIC0tPgo8ZyBpZD0ibm9kZTEiIGNsYXNzPSJub2RlIj4KPHRpdGxlPjE0MDM5OTgwNTYwMzI4MDwvdGl0bGU+CjxlbGxpcHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0id2hpdGUiIGN4PSI2MyIgY3k9Ii05MCIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iNjMiIHk9Ii04Ni4zIiBmb250LWZhbWlseT0iJ0Nhc2NhZGlhIENvZGUgUEwnLCAnSmV0QnJhaW5zIE1vbm8nLCAnSW9zZXZrYScsICdGaXJhIENvZGUnLCAnSGFjaycsIG1vbm9zcGFj
|
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|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="3ª-every-leaf-nil-is-black">3ª Every leaf (<code>nil</code>) is black.<a href="#3ª-every-leaf-nil-is-black" class="hash-link" aria-label="Direct link to 3ª-every-leaf-nil-is-black" title="Direct link to 3ª-every-leaf-nil-is-black"></a></h2>
|
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<p>Now, this rule is a funny one. What does this imply and can I interpret this in
|
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some other way? Let's go through some of the possible ways I can look at this and
|
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|
|
how would they affect the other rules and balancing.</p>
|
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|
|
<p>We will experiment with the following tree:
|
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|
<img loading="lazy" src="/assets/images/rb_light-9889570d993cf4a78a1bcccfbd76eab4.svg#gh-light-mode-only" width="899" height="539" class="img_ev3q">
|
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|
<img loading="lazy" src="/assets/images/rb_dark-2917b0f8de62597646b619102f126a53.svg#gh-dark-mode-only" width="899" height="539" class="img_ev3q"></p>
|
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<p>We should start by counting the black nodes from root to the <code>nil</code> leaves based
|
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|
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on the rules. We have multiple similar paths, so we will pick only the interesting
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|
|
ones.</p>
|
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|
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<ol>
|
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|
|
<li>What happens if we do not count the <code>nil</code> leaves?</li>
|
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|
|
<li>What happens if we consider leaves the nodes with <em>no descendants</em>, i.e. both
|
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|
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of node's children are <code>nil</code>?</li>
|
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|
|
<li>What happens if we do not count the <code>nil</code> leaves, but consider nodes with at
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least one <code>nil</code> descendant as leaves?</li>
|
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|
|
</ol>
|
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|
|
<table><thead><tr><th style="text-align:right">path</th><th style="text-align:right">black nodes</th><th style="text-align:right">1ª idea</th><th style="text-align:right">2ª idea</th><th style="text-align:right">3ª idea</th></tr></thead><tbody><tr><td style="text-align:right"><code>3 → 1 → 0 → nil</code></td><td style="text-align:right">4</td><td style="text-align:right">3</td><td style="text-align:right">4</td><td style="text-align:right">3</td></tr><tr><td style="text-align:right"><code>3 → 5 → 7 → 8 → nil</code></td><td style="text-align:right">4</td><td style="text-align:right">3</td><td style="text-align:right">-</td><td style="text-align:right">3</td></tr><tr><td style="text-align:right"><code>3 → 5 → 7 → 8 → 9 → nil</code></td><td style="text-align:right">4</td><td style="text-align:right">3</td><td style="text-align:right">4</td><td style="text-align:right">3</td></tr></tbody></table>
|
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<p>First idea is very easy to execute and it is also very easy to argue about its
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correctness. It is correct, because we just subtract one from each of the paths.
|
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This affects <strong>all</strong> paths and therefore results in global decrease by one.</p>
|
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<p>Second idea is a bit more complicated. We count the <code>nil</code>s, so the count is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">4</span></span></span></span>
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|
as it should be. However, there is one difference. Second path no longer satisfies
|
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the condition of a <em>leaf</em>. Technically it relaxes the 5th rule, because we leave
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out some of the nodes. We should probably avoid that.</p>
|
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|
|
<div class="theme-admonition theme-admonition-caution admonition_xJq3 alert alert--warning"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>caution</div><div class="admonitionContent_BuS1"><p>With the second idea, you may also feel that we are “bending” the rules a bit,
|
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|
|
especially the definition of the “leaf” nodes.</p><p>Given the definition of the red-black tree, where <code>nil</code> is considered to be an
|
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|
|
external node, we have decided that bending it a bit just to stir a thought about
|
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|
|
it won't hurt anybody. <!-- -->😉</p></div></div>
|
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|
|
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="4ª-if-a-node-is-red-then-both-its-children-are-black">4ª If a node is red, then both its children are black.<a href="#4ª-if-a-node-is-red-then-both-its-children-are-black" class="hash-link" aria-label="Direct link to 4ª If a node is red, then both its children are black." title="Direct link to 4ª If a node is red, then both its children are black."></a></h2>
|
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|
|
<p>This rule might seem rather silly on the first look, but there are 2 important
|
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|
|
functions:</p>
|
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|
|
<ol>
|
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|
|
<li>it allows the algorithms to <em>“notice”</em> that something went wrong (i.e. the
|
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|
|
tree needs to be rebalanced), and</li>
|
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|
|
<li>it holds the balancing and height of the tree <em>“in check”</em> (with the help of
|
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|
|
the 5th rule).</li>
|
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|
|
</ol>
|
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|
|
<p>When we have a look at the algorithms that are used for fixing up the red-black
|
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|
|
tree after an insertion or deletion, we will notice that all the algorithms need
|
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is the color of the node.</p>
|
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|
|
<blockquote>
|
|||
|
|
<p>How come it is the only thing that we need?
|
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|
|
How come such naïve thing can be enough?</p>
|
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|
|
</blockquote>
|
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|
|
<p>Let's say we perform an insertion into the tree… We go with the usual and pretty
|
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|
|
primitive insertion into the binary-search tree and then, if needed, we “fix up”
|
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|
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broken invariants. <em>How can that be enough?</em> With each insertion and deletion we
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maintain the invariants, therefore if we break them with one operation, there's
|
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|
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only one path on which the invariants were <em>felled</em>. If we know that rest of the
|
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|
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tree is correct, it allows us to fix the issues just by propagating it to the
|
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root and <em>abusing</em> the siblings (which are, of course, correct red-black
|
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subtrees) to fix or at least partially mitigate the issues and propagate them
|
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|
further.</p>
|
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|
|
<p>Let's assume that we do not enforce this rule, you can see how it breaks the
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balancing of the tree below.</p>
|
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|
<!-- -->
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<!-- -->
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<div class="tabs-container tabList__CuJ"><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_LNqP tabs__item--active">Enforcing this rule</li><li role="tab" tabindex="-1" aria-selected="false" class="tabs__item tabItem_LNqP">Omitting this rule</li></ul><div class="margin-top--md"><div role="tabpanel" class="tabItem_Ymn6"><p><img loading="lazy" src="data:image/svg+xml;base64,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<img loading="lazy" src="data:image/svg+xml;base64,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<img loading="lazy" src="/assets/images/incorrect_dark-d9c04aed74f7d364c3c3b1855b769ab0.svg#gh-dark-mode-only" width="803" height="443" class="img_ev3q"></p></div></div></div>
|
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<p>We can create a <strong>big</strong> subtree with only red nodes and <strong>even</strong> when keeping
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the rest of the rules maintained, it will break the time complexity. It stops us
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from “hacking” the black height requirement laid by the 5th rule.</p>
|
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="5ª-for-each-node-all-simple-paths-from-the-node-to-descendant-leaves-contain-the-same-number-of-black-nodes">5ª For each node, all simple paths from the node to descendant leaves contain the same number of black nodes.<a href="#5ª-for-each-node-all-simple-paths-from-the-node-to-descendant-leaves-contain-the-same-number-of-black-nodes" class="hash-link" aria-label="Direct link to 5ª For each node, all simple paths from the node to descendant leaves contain the same number of black nodes." title="Direct link to 5ª For each node, all simple paths from the node to descendant leaves contain the same number of black nodes."></a></h2>
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<p>As it was mentioned, with the 4th rule they hold the balancing of the red-black
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tree.</p>
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<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_BuS1"><p>An important observation here is the fact that the red-black tree is a
|
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<strong>height</strong>-balanced tree.</p></div></div>
|
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<p>Enforcing this rule (together with the 4th rule) keeps the tree balanced:</p>
|
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<ol>
|
|||
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<li>4th rule makes sure we can't “hack” this requirement.</li>
|
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|
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<li>This rule ensures that we have “similar”<sup><a href="#user-content-fn-2" id="user-content-fnref-2" data-footnote-ref="true" aria-describedby="footnote-label">2</a></sup> length to each of the leaves.</li>
|
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|
|
</ol>
|
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|
|
<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>AVL tree</div><div class="admonitionContent_BuS1"><p>You might have heard about an <em>AVL tree</em> before. It is the first self-balanced
|
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|
|
tree to be ever introduced and works in a very similar nature as the red-black
|
|||
|
|
tree, the only difference is that it does not deal with the <em>black height</em>, but
|
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|
|
the height in general.</p><p>If you were to compare AVL with the red-black tree, you can say that AVL is much
|
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|
|
more strict while red-black tree can still maintain the same asymptotic time
|
|||
|
|
complexity for the operations, but having more relaxed rules.</p></div></div>
|
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|
|
<section data-footnotes="true" class="footnotes"><h2 class="anchor anchorWithStickyNavbar_LWe7 sr-only" id="footnote-label">Footnotes<a href="#footnote-label" class="hash-link" aria-label="Direct link to Footnotes" title="Direct link to Footnotes"></a></h2>
|
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|
|
<ol>
|
|||
|
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<li id="user-content-fn-1">
|
|||
|
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<p>CORMEN, Thomas. Introduction to algorithms. Cambridge, Mass: MIT Press, 2009. isbn 9780262033848. <a href="#user-content-fnref-1" data-footnote-backref="" aria-label="Back to reference 1" class="data-footnote-backref">↩</a></p>
|
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</li>
|
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|
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<li id="user-content-fn-2">
|
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|
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<p>red nodes still exist <a href="#user-content-fnref-2" data-footnote-backref="" aria-label="Back to reference 2" class="data-footnote-backref">↩</a></p>
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|
</li>
|
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</ol>
|
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</section></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-tags-row row margin-bottom--sm"><div class="col"><b>Tags:</b><ul class="tags_jXut padding--none margin-left--sm"><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/red-black-trees/">red-black trees</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/balanced-trees/">balanced trees</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://github.com/mfocko/blog/tree/main/algorithms/08-rb-trees/2023-06-10-rules.md" target="_blank" rel="noopener noreferrer" class="theme-edit-this-page"><svg fill="currentColor" height="20" width="20" viewBox="0 0 40 40" class="iconEdit_Z9Sw" aria-hidden="true"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="col lastUpdated_vwxv"><span class="theme-last-updated">Last updated<!-- --> on <b><time datetime="2023-06-10T00:00:00.000Z">Jun 10, 2023</time></b></span></div></div></footer></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages"><a class="pagination-nav__link pagination-nav__link--prev" href="/algorithms/rb-trees/applications/"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Použití červeno-černých stromů</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/algorithms/category/graphs/"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Graphs</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#introduction" class="table-of-contents__link toc-highlight">Introduction</a></li><li><a href="#1ª-every-node-is-either-red-or-black" class="table-of-contents__link toc-highlight">1ª Every node is either red or black.</a><ul><li><a href="#do-i-really-need-the-nodes-to-be-explicitly-colored" class="table-of-contents__link toc-highlight">Do I really need the nodes to be explicitly colored?</a></li></ul></li><li><a href="#2ª-the-root-is-black" class="table-of-contents__link toc-highlight">2ª The root is black.</a></li><li><a href="#3ª-every-leaf-nil-is-black" class="table-of-contents__link toc-highlight">3ª Every leaf (<code>nil</code>) is black.</a></li><li><a href="#4ª-if-a-node-is-red-then-both-its-children-are-black" class="table-of-contents__link toc-highlight">4ª If a node is red, then both its children are black.</a></li><li><a href="#5ª-for-each-node-all-simple-paths-from-the-node-to-descendant-leaves-contain-the-same-number-of-black-nodes" class="table-of-contents__link toc-highlight">5ª For each node, all simple paths from the node to descendant leaves contain the same number of black nodes.</a></li></ul></div></div></div></div></main></div></div></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">Git</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__item"><a href="https://gitlab.com/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitLab<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 2.828 6.977-7.07 4.125 4.172v-11z"></path></svg></a></li><li class="footer__
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