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第 9 章 &nbsp; グラフ
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第 11 章 &nbsp; ソート
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<h1 id="76">7.6 &nbsp; まとめ<a class="headerlink" href="#76" title="この見出しへのリンク">&para;</a></h1>
<h3 id="1">1. &nbsp; 要点の振り返り<a class="headerlink" href="#1" title="この見出しへのリンク">&para;</a></h3>
<ul>
<li>二分木は「一つを二つに分ける」という分割統治の論理を表す非線形データ構造です。各ノードには値と、左右の子をそれぞれ指す二つのポインタがあります。</li>
<li>あるノードの左(右)の子とその下のノードが作る木を、そのノードの左(右)部分木と呼びます。</li>
<li>関連する用語には、根ノード、葉ノード、レベル、次数、辺、高さ、深さなどがあります。</li>
<li>初期化、ノードの挿入・削除の方法は、連結リストと似ています。</li>
<li>主な種類には、パーフェクト二分木、完全二分木、全二分木、平衡二分木があります。パーフェクト二分木は最も理想的な状態で、連結リストは退化した最悪の状態です。</li>
<li>二分木は配列でも表せます。ノードの値と空の位置をレベル順に並べ、親子のインデックスの対応でポインタを実現します。</li>
<li>レベル順走査は幅優先探索の一種で、一周ずつ外へ広がるように層ごとに進み、通常はキューで実装します。</li>
<li>行きがけ順、通りがけ順、帰りがけ順は深さ優先探索で、行き止まりまで進んでから戻って続ける方法です。通常は再帰で実装します。</li>
<li>二分探索木は要素を効率よく探せるデータ構造で、探索、挿入、削除の時間計算量はすべて <span class="arithmatex">\(O(\log n)\)</span> です。連結リストに退化すると、いずれも <span class="arithmatex">\(O(n)\)</span> に悪化します。</li>
<li>AVL 木は平衡二分探索木とも呼ばれ、回転によって、挿入と削除を続けても平衡を保ちます。</li>
<li>AVL 木の回転には、右回転、左回転、右回転してから左回転、左回転してから右回転があります。挿入や削除の後、下から上へ回転して平衡を取り戻します。</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="この見出しへのリンク">&para;</a></h3>
<p><strong>Q</strong>:ノードが一つだけの二分木では、木の高さも根の深さも <span class="arithmatex">\(0\)</span> ですか?</p>
<p>はい。通常は、通る辺の数で高さと深さを定義するためです。</p>
<p><strong>Q</strong>:二分木の挿入・削除は複数の操作を組み合わせるとありますが、どういう意味ですか? リソースの子ノードのリソースを解放することと考えてよいですか?</p>
<p>二分探索木を例にすると、削除には三つの場合があり、それぞれで複数のノード操作が必要です。</p>
<p><strong>Q</strong>:二分木の DFS には、なぜ行きがけ順、通りがけ順、帰りがけ順があり、それぞれ何に使いますか?</p>
<p>配列を順方向や逆方向に走査するのと同様、これらは特定の順序で結果を得るための三つの走査方法です。例えば二分探索木では、<code>左の子の値 &lt; 根の値 &lt; 右の子の値</code> が成り立つため、「左 <span class="arithmatex">\(\rightarrow\)</span> 根 <span class="arithmatex">\(\rightarrow\)</span> 右」の順で走査すれば、整列済みのノード列を得られます。</p>
<p><strong>Q</strong>:右回転は不均衡な <code>node</code>、<code>child</code>、<code>grand_child</code> の関係を処理しますが、<code>node</code> と親のつながりは更新しなくてよいのですか? 切れてしまいませんか?</p>
<p>再帰の視点から考える必要があります。<code>right_rotate(root)</code> は部分木の根を受け取り、最後に <code>return child</code> で回転後の根を返します。部分木の根と親の接続は、この関数から戻った後に行うので、右回転の処理範囲には含まれません。</p>
<p><strong>Q</strong>:C++ で関数を <code>private</code> と <code>public</code> に分ける基準は何ですか? <code>height()</code> は <code>public</code>、<code>updateHeight()</code> は <code>private</code> なのはなぜですか?</p>
<p>主に利用範囲で決めます。クラス内部だけで使うなら <code>private</code> にします。例えば <code>updateHeight()</code> は挿入や削除の一段階なので、利用者が単独で呼ぶ意味はありません。一方、<code>height()</code> は <code>vector.size()</code> のように高さを取得するものなので、使えるように <code>public</code> にします。</p>
<p><strong>Q</strong>:入力データから二分探索木をどう構築しますか? 根の選び方は重要ですか?</p>
<p>はい。構築方法は、二分探索木のコードの <code>build_tree()</code> にあります。通常、入力をソートして中点の要素を根とし、左右の部分木を再帰的に構築します。こうすることで、平衡性をできるだけ高く保てます。</p>
<p><strong>Q</strong>:Java の文字列比較には、必ず <code>equals()</code> を使いますか?</p>
<p>Java では、基本データ型の <code>==</code> は二つの変数の値が等しいかを比較します。参照型では、二つの方法の仕組みが異なります。</p>
<ul>
<li><code>==</code>:二つの変数が同じオブジェクト、つまり同じメモリ位置を指すかを比較します。</li>
<li><code>equals()</code>:二つのオブジェクトの値が等しいかを比較します。</li>
</ul>
<p>値を比較するなら <code>equals()</code> を使うべきです。ただし、<code>String a = "hi"; String b = "hi";</code> で初期化した文字列は文字列定数プールに格納され、同じオブジェクトを指すため、<code>a == b</code> でも内容を比較できます。</p>
<p><strong>Q</strong>:幅優先走査で最下層に到達する直前、キューのノード数は <span class="arithmatex">\(2^h\)</span> ですか?</p>
<p>はい。例えば高さ <span class="arithmatex">\(h = 2\)</span> のパーフェクト二分木なら、総ノード数は <span class="arithmatex">\(n = 7\)</span> で、最下層のノード数は <span class="arithmatex">\(4 = 2^h = (n + 1) / 2\)</span> です。</p>
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