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10.1 &nbsp; 이진 탐색
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10.2 &nbsp; 이진 탐색으로 삽입 위치 찾기
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10.3 &nbsp; 이진 탐색으로 경계 찾기
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10.4 &nbsp; 해시를 이용한 최적화 전략
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제 11 장 &nbsp; 정렬
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11.2 &nbsp; 선택 정렬
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11.3 &nbsp; 버블 정렬
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11.4 &nbsp; 삽입 정렬
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11.5 &nbsp; 퀵 정렬
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11.6 &nbsp; 병합 정렬
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11.8 &nbsp; 버킷 정렬
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11.9 &nbsp; 계수 정렬
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11.10 &nbsp; 기수 정렬
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12.1 &nbsp; 분할 정복 알고리즘
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12.2 &nbsp; 분할 정복 탐색 전략
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12.4 &nbsp; 하노이의 탑 문제
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12.5 &nbsp; 요약
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제 13 장 &nbsp; 백트래킹
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13.4 &nbsp; n-퀸 문제
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<h1 id="77">7.7 &nbsp; 연습 문제<a class="headerlink" href="#77" title="고유 링크">&para;</a></h1>
<h2 id="771">7.7.1 &nbsp; 개념 확인<a class="headerlink" href="#771" title="고유 링크">&para;</a></h2>
<h3 id="1">1. &nbsp; 완전 이진 트리, 정 이진 트리, 포화 이진 트리<a class="headerlink" href="#1" title="고유 링크">&para;</a></h3>
<p>다음 두 배열은 이진 트리를 레벨 순서로 나타낸 것이다. <code>None</code>은 빈자리를 뜻한다.</p>
<ul>
<li>트리 A: <code>[1, 2, 3, 4, 5, 6]</code></li>
<li>트리 B: <code>[1, 2, 3, None, None, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>어느 트리가 완전 이진 트리인가?</li>
<li>어느 트리가 모든 비리프 노드에 자식이 두 개씩 있는 정 이진 트리인가?</li>
<li>두 트리 중 포화 이진 트리가 있는가? 각각 이유를 설명하라.</li>
</ol>
<details class="success">
<summary>참고 답안</summary>
<ol>
<li>
<p>트리 A는 완전 이진 트리다. 맨 아래 레벨만 가득 차지 않았고, 노드가 왼쪽부터 연속해서 배치되어 있다.
트리 B는 맨 아래 레벨의 왼쪽에 빈자리가 있는데 오른쪽에 노드가 있으므로 완전 이진 트리가 아니다.</p>
</li>
<li>
<p>트리 B는 정 이진 트리다. 노드 1과 노드 3은 자식이 각각 두 개이고, 나머지는 모두 리프 노드다.
트리 A의 노드 3에는 왼쪽 자식 6만 있으므로 정 이진 트리가 아니다.</p>
</li>
<li>
<p>두 트리 모두 맨 아래 레벨이 완전히 채워져 있지 않으므로 포화 이진 트리가 아니다.</p>
</li>
</ol>
</details>
<h3 id="2">2. &nbsp; 같은 트리의 세 가지 순회 순서<a class="headerlink" href="#2" title="고유 링크">&para;</a></h3>
<p>배열 <code>[1, 2, 3, 4, 5, 6, 7]</code>을 레벨 순서로 완전 이진 트리에 저장한다.</p>
<!-- numbered-subquestions -->
<ol>
<li>이 트리를 그려 보자.</li>
<li>전위, 중위, 후위 순회 결과를 각각 적어 보자.</li>
<li>중위 순회 결과에서 루트 노드 1의 왼쪽과 오른쪽에 있는 두 수열은 각각 트리의 어느 부분에 대응하는가?</li>
</ol>
<details class="success">
<summary>참고 답안</summary>
<ol>
<li>
<p>트리는 다음과 같다.</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a> 1
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> / \
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> 2 3
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> / \ / \
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> 4 5 6 7
</code></pre></div>
</li>
<li>
<p>전위 순회는 <code>1, 2, 4, 5, 3, 6, 7</code>이다.
중위 순회는 <code>4, 2, 5, 1, 6, 3, 7</code>이다.
후위 순회는 <code>4, 5, 2, 6, 7, 3, 1</code>이다.</p>
</li>
<li>
<p>루트 노드 1의 왼쪽에 있는 <code>4, 2, 5</code>는 왼쪽 서브트리의 중위 순회 결과다.
오른쪽의 <code>6, 3, 7</code>은 오른쪽 서브트리의 중위 순회 결과다.</p>
</li>
</ol>
</details>
<h3 id="3">3. &nbsp; 두 이진 탐색 트리 비교하기<a class="headerlink" href="#3" title="고유 링크">&para;</a></h3>
<p>다음 두 수열을 각각 왼쪽부터 빈 이진 탐색 트리에 차례대로 삽입한다.</p>
<ul>
<li>수열 A: <code>[4, 2, 6, 1, 3, 5, 7]</code></li>
<li>수열 B: <code>[1, 2, 3, 4, 5, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>숫자 7을 찾을 때 거치는 노드를 각각 적어 보자.</li>
<li>높이를 루트에서 가장 먼 리프까지 지나는 간선 수로 계산할 때, 두 트리의 높이는 각각 얼마인가?</li>
<li>앞선 두 질문을 바탕으로 두 트리에서 7을 찾는 효율성이 같은지 판단하라. 트리의 모양과 탐색 경로를 바탕으로 이유를 설명하라.</li>
</ol>
<details class="success">
<summary>참고 답안</summary>
<ol>
<li>
<p>수열 A로 만든 트리의 탐색 경로는 <code>4 → 6 → 7</code>이다.
수열 B로 만든 트리의 탐색 경로는 <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>이다.</p>
</li>
<li>
<p>첫 번째 트리는 모든 레벨이 가득 차 있고 높이는 2다. 두 번째 트리는 오른쪽 자식만 있으며 높이는 6이다.</p>
</li>
<li>
<p>두 트리에서 7을 찾는 효율성은 다르다. 삽입 순서가 이진 탐색 트리의 모양과 높이를 바꾸기 때문이다. 첫 번째 트리는 노드 3개만 거치지만,
두 번째 트리는 노드 7개를 차례대로 거친다. 트리가 높을수록 최악의 경우 경로를 따라 비교해야 하는 노드가 많아진다.</p>
</li>
</ol>
</details>
<h2 id="772">7.7.2 &nbsp; 프로그래밍 연습<a class="headerlink" href="#772" title="고유 링크">&para;</a></h2>
<h3 id="1_1">1. &nbsp; 이진 트리의 최대 깊이<a class="headerlink" href="#1_1" title="고유 링크">&para;</a></h3>
<p>이진 트리의 루트 노드 <code>root</code>가 주어진다. 각 노드는 정숫값과 왼쪽 및 오른쪽 자식 노드에 대한 참조를 가진다.</p>
<p>루트에서 가장 먼 리프까지 거치는 <strong>노드 수</strong>를 이진 트리의 최대 깊이라고 정의한다. 이 트리의 최대 깊이를 반환하라. 빈 트리의 최대 깊이는 0이다.
재귀를 사용하여 해결하라.</p>
<details class="tip">
<summary>풀이 힌트</summary>
<ol>
<li>이 문제에서는 노드 수로 깊이를 계산한다. 루트 하나만 있는 트리의 최대 깊이는 1이다.</li>
<li>재귀 함수가 현재 노드를 루트로 하는 서브트리의 최대 깊이를 반환하도록 한다.</li>
<li>빈 노드는 0을 반환하고, 비어 있지 않은 노드는 max(depth(left), depth(right)) + 1을 반환한다.</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/maximum-depth-of-binary-tree/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
<h3 id="2_1">2. &nbsp; 이진 트리를 레벨 순서로 순회하기<a class="headerlink" href="#2_1" title="고유 링크">&para;</a></h3>
<p>이진 트리의 루트 노드 <code>root</code>가 주어진다. 큐를 사용하여 위에서 아래로 한 레벨씩 모든 노드를 방문하라. 같은 레벨에서는 왼쪽부터 오른쪽으로 방문한다.</p>
<p>2차원 배열을 반환하라. 첫 번째 하위 배열에는 루트가 있는 레벨의 노드 값을, 두 번째에는 다음 레벨의 노드 값을 저장하며 이 방식을 반복한다.
빈 트리라면 빈 배열을 반환하라.</p>
<details class="tip">
<summary>풀이 힌트</summary>
<ol>
<li>레벨 순서 순회에서는 먼저 들어온 노드를 먼저 방문해야 하므로 큐를 사용한다.</li>
<li>각 반복이 시작될 때 큐에 있는 노드들은 정확히 같은 레벨에 속한다.</li>
<li>큐의 길이를 먼저 기록한 뒤 그 수만큼 노드를 꺼내고 각 노드의 자식을 넣는다.</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/binary-tree-level-order-traversal/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
<h3 id="3-k">3. &nbsp; 이진 탐색 트리에서 k번째로 작은 원소<a class="headerlink" href="#3-k" title="고유 링크">&para;</a></h3>
<p>이진 탐색 트리에 총 <code>n</code>개의 노드가 있으며, 노드 값은 모두 다르다.
모든 노드 값을 작은 것부터 나열하고 위치에 1부터 번호를 붙인다.</p>
<p>루트 노드 <code>root</code>와 <code>1 &lt;= k &lt;= n</code>을 만족하는 정수 <code>k</code>가 주어질 때, <code>k</code>번째 노드 값을 반환하라.
모든 값을 미리 모으지 말고 중위 순회 도중에 바로 답을 찾아라.</p>
<details class="tip">
<summary>풀이 힌트</summary>
<ol>
<li>이진 탐색 트리를 중위 순회하면 노드 값을 작은 것부터 방문한다.</li>
<li>중위 순회는 왼쪽 서브트리, 현재 노드, 오른쪽 서브트리 순서로 처리한다. 현재 노드를 방문할 때 개수를 1 늘린다.</li>
<li>개수가 처음 k가 되었을 때의 노드 값이 답이다. 이후에는 순회를 계속할 필요가 없다.</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/kth-smallest-element-in-a-bst/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
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