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数组，结构体，二叉树
引入
有时候我们会遇到一些大规模的区间查找和区间修改问题，比如让你维护一个 10^5 长度的数列，要求操作有区间求和、区间加（区间每个数加上一个值），让你在一秒内完成 10^5 次操作。 暴力是肯定不行的，数据范围太大，会超时。 所以我们就有一种专门解决大范围区间修改查询的数据结构：线段树。">
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数组，结构体，二叉树
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有时候我们会遇到一些大规模的区间查找和区间修改问题，比如让你维护一个 10^5 长度的数列，要求操作有区间求和、区间加（区间每个数加上一个值），让你在一秒内完成 10^5 次操作。 暴力是肯定不行的，数据范围太大，会超时。 所以我们就有一种专门解决大范围区间修改查询的数据结构：线段树。">
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- <p>数组，结构体，二叉树</p>
- <h3 id="引入">引入</h3>
- <p>有时候我们会遇到一些大规模的区间查找和区间修改问题，比如让你维护一个 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span> 长度的数列，要求操作有区间求和、区间加（区间每个数加上一个值），让你在一秒内完成 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span> 次操作。 暴力是肯定不行的，数据范围太大，会超时。 所以我们就有一种专门解决大范围区间修改查询的数据结构：线段树。</p>
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- <h3 id="线段树">线段树</h3>
- <p>线段树本质上是把整个数列拆分了，用一个一个区间来表示。 比如有 <span class="katex"><span class="katex-mathml"><math><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 mathdefault">n</span></span></span></span>个节点 根节点代表整个数列的区间<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">]</span></span></span></span>， 根节点的两个子节点代表根区间二分成两部分，也就是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mi mathvariant="normal">mid</mi><mo>⁡</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,\operatorname{mid}]</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">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mord mathrm">m</span><span class="mord mathrm">i</span><span class="mord mathrm">d</span></span><span class="mclose">]</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi mathvariant="normal">mid</mi><mo>⁡</mo><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\operatorname{mid},n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mop"><span class="mord mathrm">m</span><span class="mord mathrm">i</span><span class="mord mathrm">d</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">]</span></span></span></span>。 而子节点也是一棵线段树。 一直往下眼神，叶子结点就代表着单一位置<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,a]</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 mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">a</span><span class="mclose">]</span></span></span></span></p>
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