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7.6 &nbsp; 小結
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第 8 章 &nbsp; 堆積
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第 9 章 &nbsp; 圖
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第 9 章 &nbsp; 圖
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9.2 &nbsp; 圖基礎操作
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第 10 章 &nbsp; 搜尋
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10.1 &nbsp; 二分搜尋
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10.2 &nbsp; 二分搜尋插入點
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10.3 &nbsp; 二分搜尋邊界
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10.4 &nbsp; 雜湊最佳化策略
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10.5 &nbsp; 重識搜尋演算法
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第 11 章 &nbsp; 排序
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11.1 &nbsp; 排序演算法
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11.2 &nbsp; 選擇排序
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11.3 &nbsp; 泡沫排序
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11.4 &nbsp; 插入排序
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11.5 &nbsp; 快速排序
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11.6 &nbsp; 合併排序
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11.7 &nbsp; 堆積排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 計數排序
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11.10 &nbsp; 基數排序
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11.11 &nbsp; 小結
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治演算法
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12.3 &nbsp; 構建樹問題
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12.4 &nbsp; 河內塔問題
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第 13 章 &nbsp; 回溯
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13.2 &nbsp; 全排列問題
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13.3 &nbsp; 子集和問題
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13.4 &nbsp; N 皇后問題
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第 14 章 &nbsp; 動態規劃
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14.1 &nbsp; 初探動態規劃
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14.2 &nbsp; DP 問題特性
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14.3 &nbsp; DP 解題思路
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14.4 &nbsp; 0-1 背包問題
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14.5 &nbsp; 完全背包問題
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<h1 id="76">7.6 &nbsp; 小結<a class="headerlink" href="#76" title="Permanent link">&para;</a></h1>
<h3 id="1">1. &nbsp; 重點回顧<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<ul>
<li>二元樹是一種非線性資料結構，體現“一分為二”的分治邏輯。每個二元樹節點包含一個值以及兩個指標，分別指向其左子節點和右子節點。</li>
<li>對於二元樹中的某個節點，其左（右）子節點及其以下形成的樹被稱為該節點的左（右）子樹。</li>
<li>二元樹的相關術語包括根節點、葉節點、層、度、邊、高度和深度等。</li>
<li>二元樹的初始化、節點插入和節點刪除操作與鏈結串列操作方法類似。</li>
<li>常見的二元樹型別有完美二元樹、完全二元樹、完滿二元樹和平衡二元樹。完美二元樹是最理想的狀態，而鏈結串列是退化後的最差狀態。</li>
<li>二元樹可以用陣列表示，方法是將節點值和空位按層序走訪順序排列，並根據父節點與子節點之間的索引對映關係來實現指標。</li>
<li>二元樹的層序走訪是一種廣度優先搜尋方法，它體現了“一圈一圈向外擴展”的逐層走訪方式，通常透過佇列來實現。</li>
<li>前序、中序、後序走訪皆屬於深度優先搜尋，它們體現了“先走到盡頭，再回溯繼續”的走訪方式，通常使用遞迴來實現。</li>
<li>二元搜尋樹是一種高效的元素查詢資料結構，其查詢、插入和刪除操作的時間複雜度均為 <span class="arithmatex">\(O(\log n)\)</span> 。當二元搜尋樹退化為鏈結串列時，各項時間複雜度會劣化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
<li>AVL 樹，也稱平衡二元搜尋樹，它透過旋轉操作確保在不斷插入和刪除節點後樹仍然保持平衡。</li>
<li>AVL 樹的旋轉操作包括右旋、左旋、先右旋再左旋、先左旋再右旋。在插入或刪除節點後，AVL 樹會從底向頂執行旋轉操作，使樹重新恢復平衡。</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>：對於只有一個節點的二元樹，樹的高度和根節點的深度都是 <span class="arithmatex">\(0\)</span> 嗎？</p>
<p>是的，因為高度和深度通常定義為“經過的邊的數量”。</p>
<p><strong>Q</strong>：二元樹中的插入與刪除一般由一套操作配合完成，這裡的“一套操作”指什麼呢？可以理解為資源的子節點的資源釋放嗎？</p>
<p>拿二元搜尋樹來舉例，刪除節點操作要分三種情況處理，其中每種情況都需要進行多個步驟的節點操作。</p>
<p><strong>Q</strong>：為什麼 DFS 走訪二元樹有前、中、後三種順序，分別有什麼用呢？</p>
<p>與順序和逆序走訪陣列類似，前序、中序、後序走訪是三種二元樹走訪方法，我們可以使用它們得到一個特定順序的走訪結果。例如在二元搜尋樹中，由於節點大小滿足 <code>左子節點值 &lt; 根節點值 &lt; 右子節點值</code> ，因此我們只要按照“左 <span class="arithmatex">\(\rightarrow\)</span> 根 <span class="arithmatex">\(\rightarrow\)</span> 右”的優先順序走訪樹，就可以獲得有序的節點序列。</p>
<p><strong>Q</strong>：右旋操作是處理失衡節點 <code>node</code>、<code>child</code>、<code>grand_child</code> 之間的關係，那 <code>node</code> 的父節點和 <code>node</code> 原來的連線不需要維護嗎？右旋操作後豈不是斷掉了？</p>
<p>我們需要從遞迴的視角來看這個問題。右旋操作 <code>right_rotate(root)</code> 傳入的是子樹的根節點，最終 <code>return child</code> 返回旋轉之後的子樹的根節點。子樹的根節點和其父節點的連線是在該函式返回後完成的，不屬於右旋操作的維護範圍。</p>
<p><strong>Q</strong>：在 C++ 中，函式被劃分到 <code>private</code> 和 <code>public</code> 中，這方面有什麼考量嗎？為什麼要將 <code>height()</code> 函式和 <code>updateHeight()</code> 函式分別放在 <code>public</code> 和 <code>private</code> 中呢？</p>
<p>主要看方法的使用範圍，如果方法只在類別內部使用，那麼就設計為 <code>private</code> 。例如，使用者單獨呼叫 <code>updateHeight()</code> 是沒有意義的，它只是插入、刪除操作中的一步。而 <code>height()</code> 是訪問節點高度，類似於 <code>vector.size()</code> ，因此設定成 <code>public</code> 以便使用。</p>
<p><strong>Q</strong>：如何從一組輸入資料構建一棵二元搜尋樹？根節點的選擇是不是很重要？</p>
<p>是的，構建樹的方法已在二元搜尋樹程式碼中的 <code>build_tree()</code> 方法中給出。至於根節點的選擇，我們通常會將輸入資料排序，然後將中點元素作為根節點，再遞迴地構建左右子樹。這樣做可以最大程度保證樹的平衡性。</p>
<p><strong>Q</strong>：在 Java 中，字串對比是否一定要用 <code>equals()</code> 方法？</p>
<p>在 Java 中，對於基本資料型別，<code>==</code> 用於對比兩個變數的值是否相等。對於引用型別，兩種符號的工作原理是不同的。</p>
<ul>
<li><code>==</code> ：用來比較兩個變數是否指向同一個物件，即它們在記憶體中的位置是否相同。</li>
<li><code>equals()</code>：用來對比兩個物件的值是否相等。</li>
</ul>
<p>因此，如果要對比值，我們應該使用 <code>equals()</code> 。然而，透過 <code>String a = "hi"; String b = "hi";</code> 初始化的字串都儲存在字串常數池中，它們指向同一個物件，因此也可以用 <code>a == b</code> 來比較兩個字串的內容。</p>
<p><strong>Q</strong>：廣度優先走訪到最底層之前，佇列中的節點數量是 <span class="arithmatex">\(2^h\)</span> 嗎？</p>
<p>是的，例如高度 <span class="arithmatex">\(h = 2\)</span> 的滿二元樹，其節點總數 <span class="arithmatex">\(n = 7\)</span> ，則底層節點數量 <span class="arithmatex">\(4 = 2^h = (n + 1) / 2\)</span> 。</p>
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