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<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
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<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
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Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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5.3 Double-Ended Queue
5.3 Deque
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Chapter 6. Hashing
Chapter 6. Hash Table
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Chapter 6. Hashing
Chapter 6. Hash Table
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7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
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7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
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8.2 Building a Heap
8.2 Heap Construction Operation
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8.3 Top-K Problem
8.3 Top-k Problem
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10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
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10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
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10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
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11.1 Sorting Algorithms
11.1 Sorting Algorithm
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12.4 Hanoi Tower Problem
12.4 Hanota Problem
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16.3 Terminology Table
16.3 Glossary
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<!-- Page content -->
<h1 id="73-array-representation-of-binary-trees">7.3 &nbsp; Array Representation of Binary Trees<a class="headerlink" href="#73-array-representation-of-binary-trees" title="Permanent link">&para;</a></h1>
<p>Under the linked list representation, the storage unit of a binary tree is a node <code>TreeNode</code>, and nodes are connected by pointers. The previous section introduced the basic operations of binary trees under the linked list representation.</p>
<p>In the linked-list representation, the storage unit of a binary tree is a node <code>TreeNode</code>, and nodes are connected by pointers. The previous section introduced the basic operations of binary trees in this representation.</p>
<p>So, can we use an array to represent a binary tree? The answer is yes.</p>
<h2 id="731-representing-perfect-binary-trees">7.3.1 &nbsp; Representing Perfect Binary Trees<a class="headerlink" href="#731-representing-perfect-binary-trees" title="Permanent link">&para;</a></h2>
<p>Let's analyze a simple case first. Given a perfect binary tree, we store all nodes in an array according to the order of level-order traversal, where each node corresponds to a unique array index.</p>
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<p><img alt="Level-order traversal sequence corresponds to multiple binary tree possibilities" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_without_empty.png" /></p>
<p align="center"> Figure 7-13 &nbsp; Level-order traversal sequence corresponds to multiple binary tree possibilities </p>
<p>To solve this problem, <strong>we can consider explicitly writing out all <code>None</code> values in the level-order traversal sequence</strong>. As shown in Figure 7-14, after this treatment, the level-order traversal sequence can uniquely represent a binary tree. Example code is as follows:</p>
<p>To solve this problem, <strong>we can explicitly write out all <code>None</code> values in the level-order traversal sequence</strong>. As shown in Figure 7-14, once we do this, the level-order traversal sequence can uniquely represent a binary tree. Example code is as follows:</p>
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<p><img alt="Array representation of any type of binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_with_empty.png" /></p>
<p align="center"> Figure 7-14 &nbsp; Array representation of any type of binary tree </p>
<p><img alt="Array representation of an arbitrary binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_with_empty.png" /></p>
<p align="center"> Figure 7-14 &nbsp; Array representation of an arbitrary binary tree </p>
<p>It's worth noting that <strong>complete binary trees are very well-suited for array representation</strong>. Recalling the definition of a complete binary tree, <code>None</code> only appears at the bottom level and towards the right, <strong>meaning all <code>None</code> values must appear at the end of the level-order traversal sequence</strong>.</p>
<p>This means that when using an array to represent a complete binary tree, it's possible to omit storing all <code>None</code> values, which is very convenient. Figure 7-15 gives an example.</p>
<p><img alt="Array representation of a complete binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" /></p>
<p align="center"> Figure 7-15 &nbsp; Array representation of a complete binary tree </p>
<p>The following code implements a binary tree based on array representation, including the following operations:</p>
<p>The following code implements a binary tree using an array representation, including the following operations:</p>
<ul>
<li>Given a certain node, obtain its value, left (right) child node, and parent node.</li>
<li>Given a node, obtain its value, left (right) child node, and parent node.</li>
<li>Obtain the preorder, inorder, postorder, and level-order traversal sequences.</li>
</ul>
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