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第 8 章 &nbsp; ヒープ
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8.2 &nbsp; ヒープ構築
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第 9 章 &nbsp; グラフ
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第 9 章 &nbsp; グラフ
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第 10 章 &nbsp; 探索
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10.1 &nbsp; 二分探索
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10.2 &nbsp; 二分探索の挿入位置
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10.3 &nbsp; 二分探索の境界
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10.4 &nbsp; ハッシュによる最適化戦略
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10.5 &nbsp; 探索アルゴリズム再考
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第 11 章 &nbsp; ソート
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11.5 &nbsp; クイックソート
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11.6 &nbsp; マージソート
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11.7 &nbsp; ヒープソート
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11.8 &nbsp; バケットソート
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11.9 &nbsp; 計数ソート
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11.10 &nbsp; 基数ソート
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11.11 &nbsp; まとめ
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第 12 章 &nbsp; 分割統治
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12.1 &nbsp; 分割統治法
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12.2 &nbsp; 分割統治探索戦略
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12.3 &nbsp; 二分木の構築問題
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12.4 &nbsp; ハノイの塔の問題
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12.5 &nbsp; まとめ
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第 13 章 &nbsp; バックトラッキング
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13.1 &nbsp; バックトラッキングアルゴリズム
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13.2 &nbsp; 全順列問題
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13.3 &nbsp; 部分和問題
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13.4 &nbsp; n クイーン問題
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第 14 章 &nbsp; 動的計画法
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14.1 &nbsp; 動的計画法入門
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14.2 &nbsp; 動的計画法の問題特性
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14.4 &nbsp; 0-1 ナップサック問題
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<h1 id="77">7.7 &nbsp; 演習<a class="headerlink" href="#77" title="Permanent link">&para;</a></h1>
<h2 id="771">7.7.1 &nbsp; 確認問題<a class="headerlink" href="#771" title="Permanent link">&para;</a></h2>
<h3 id="1">1. &nbsp; 完全二分木・充満二分木・充足二分木<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>次の 2 つの配列は二分木をレベル順に表しており、<code>None</code> は空き位置を表します。</p>
<ul>
<li>木 A：<code>[1, 2, 3, 4, 5, 6]</code></li>
<li>木 B：<code>[1, 2, 3, None, None, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>どちらが完全二分木ですか？</li>
<li>どちらが充満二分木、つまりすべての非葉ノードが 2 つの子ノードを持つ二分木ですか？</li>
<li>この 2 つに充足二分木はありますか？それぞれ理由を説明してください。</li>
</ol>
<details class="success">
<summary>解答</summary>
<ol>
<li>
<p>木 A は完全二分木です。最下層だけがすべて埋まっておらず、ノードが左から右へ隙間なく並んでいます。
木 B は完全二分木ではありません。最下層の左側に空き位置がある一方で、その右側にはノードがあるためです。</p>
</li>
<li>
<p>木 B は充満二分木です。ノード 1 とノード 3 はそれぞれ 2 つの子ノードを持ち、それ以外のノードはすべて葉ノードです。
木 A は充満二分木ではありません。ノード 3 が左子ノード 6 だけを持つためです。</p>
</li>
<li>
<p>どちらも充足二分木ではありません。どちらの木も最下層がすべて埋まっていないためです。</p>
</li>
</ol>
</details>
<h3 id="2-3">2. &nbsp; 同じ木を走査する 3 つの順序<a class="headerlink" href="#2-3" title="Permanent link">&para;</a></h3>
<p>配列 <code>[1, 2, 3, 4, 5, 6, 7]</code> をレベル順に完全二分木へ格納します。</p>
<!-- numbered-subquestions -->
<ol>
<li>この木を描いてください。</li>
<li>先行順・中間順・後行順走査の列をそれぞれ書いてください。</li>
<li>中間順走査の列で、根ノード 1 の左側と右側にある 2 つの列は、木のどの部分にそれぞれ対応しますか？</li>
</ol>
<details class="success">
<summary>解答</summary>
<ol>
<li>
<p>この木は次のとおりです。</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>先行順走査は <code>1, 2, 4, 5, 3, 6, 7</code>、
中間順走査は <code>4, 2, 5, 1, 6, 3, 7</code>、
後行順走査は <code>4, 5, 2, 6, 7, 3, 1</code> です。</p>
</li>
<li>
<p>根ノード 1 の左側にある <code>4, 2, 5</code> は、左部分木の中間順走査の列です。
右側にある <code>6, 3, 7</code> は、右部分木の中間順走査の列です。</p>
</li>
</ol>
</details>
<h3 id="3-2">3. &nbsp; 2 つの二分探索木を比較する<a class="headerlink" href="#3-2" title="Permanent link">&para;</a></h3>
<p>次の 2 つの列を、左から右へ順に空の二分探索木へ挿入します。</p>
<ul>
<li>列 A：<code>[4, 2, 6, 1, 3, 5, 7]</code></li>
<li>列 B：<code>[1, 2, 3, 4, 5, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>それぞれの木で 7 を探索するときに通るノードを書いてください。</li>
<li>高さを根ノードから最も遠い葉ノードまでに通る辺の数とすると、2 つの木の高さはそれぞれいくつですか？</li>
<li>問い 1、2 の結果から、2 つの木で 7 を探索する効率は同じだと考えられますか？木の形と探索経路をもとに理由を説明してください。</li>
</ol>
<details class="success">
<summary>解答</summary>
<ol>
<li>
<p>列 A から作った木では、探索経路は <code>4 → 6 → 7</code> です。
列 B から作った木では、探索経路は <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code> です。</p>
</li>
<li>
<p>1 つ目の木はすべての層が埋まっているため、高さは 2 です。2 つ目の木では各ノードが右の子だけを持ち、高さは 6 です。</p>
</li>
<li>
<p>2 つの木で 7 を探索する効率は異なります。挿入順序によって二分探索木の形と高さが変わります。1 つ目の木で 7 を探索するときに通るノードは 3 つだけですが、
2 つ目の木では 7 つのノードを順に通ります。木が高いほど、最悪の場合に経路上で比較するノードが多くなります。</p>
</li>
</ol>
</details>
<h2 id="772">7.7.2 &nbsp; プログラミング演習<a class="headerlink" href="#772" title="Permanent link">&para;</a></h2>
<h3 id="1_1">1. &nbsp; 二分木の最大の深さ<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
<p>二分木の根ノード <code>root</code> が与えられます。各ノードには、整数値と左右の子ノードを指す参照が含まれています。</p>
<p>根ノードから最も遠い葉ノードまでに通る**ノード数**を、二分木の最大の深さとします。この木の最大の深さを返してください。空の木の最大の深さは 0 です。
再帰を使って求めてください。</p>
<details class="tip">
<summary>解法のヒント</summary>
<ol>
<li>この問題ではノード数で深さを数えます。根ノードだけがある場合、最大の深さは 1 です</li>
<li>再帰関数が、現在のノードを根とする部分木の最大の深さを返すようにします</li>
<li>空のノードは 0 を返し、空でないノードは max(depth(left), depth(right)) + 1 を返します</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">2. &nbsp; 二分木のレベル順走査<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>二分木の根ノード <code>root</code> が与えられます。キューを使い、上から下へ層ごとにすべてのノードを訪問してください。同じ層では、左から右の順に訪問します。</p>
<p>二次元配列を返してください。1 つ目の部分配列には根ノードがある層のノード値、2 つ目の部分配列には次の層のノード値を保存し、以降も同様に続けます。
二分木が空の場合は、空の配列を返してください。</p>
<details class="tip">
<summary>解法のヒント</summary>
<ol>
<li>レベル順走査では先に入ったノードを先に訪問するため、キューを使います</li>
<li>各回の開始時にキューに入っているノードは、ちょうど同じ層に属しています</li>
<li>先にキューの長さを記録し、その個数のノードを取り出して、それぞれの子ノードを追加します</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; 二分探索木で k 番目に小さい要素<a class="headerlink" href="#3-k" title="Permanent link">&para;</a></h3>
<p>二分探索木には <code>n</code> 個のノードがあり、すべてのノード値は互いに異なります。
すべてのノード値を小さい順に並べたときの位置には、1 から番号を付けます。</p>
<p>根ノード <code>root</code> と、<code>1 &lt;= k &lt;= n</code> を満たす整数 <code>k</code> が与えられます。<code>k</code> 番目のノード値を返してください。
すべてのノード値を先に集めるのではなく、中間順走査の途中で答えを直接探してください。</p>
<details class="tip">
<summary>解法のヒント</summary>
<ol>
<li>二分探索木を中間順走査すると、ノード値を小さい順に訪問できます</li>
<li>中間順走査では、左部分木、現在のノード、右部分木の順に処理します。現在のノードを訪問するとき、カウントに 1 を足します</li>
<li>カウントが初めて k になったとき、現在のノード値が答えです。それ以降は走査を続ける必要がありません</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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