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<p><a class="glightbox" href="../avl_tree.assets/avltree_degradation_from_inserting_node.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Degradation of an AVL tree after inserting nodes" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></a></p>
<p align="center"> Figure 7-25 &nbsp; Degradation of an AVL tree after inserting nodes </p>
<p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the "AVL Tree" in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at <span class="arithmatex">\(O(\log n)\)</span> level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>
<p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the <u>AVL Tree</u> in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at <span class="arithmatex">\(O(\log n)\)</span> level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.</p>
<h2 id="751-common-terminology-in-avl-trees">7.5.1 &nbsp; Common terminology in AVL trees<a class="headerlink" href="#751-common-terminology-in-avl-trees" title="Permanent link">&para;</a></h2>
<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a "balanced binary search tree".</p>
<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a <u>balanced binary search tree</u>.</p>
<h3 id="1-node-height">1. &nbsp; Node height<a class="headerlink" href="#1-node-height" title="Permanent link">&para;</a></h3>
<p>Since the operations related to AVL trees require obtaining node heights, we need to add a <code>height</code> variable to the node class:</p>
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<h3 id="2-node-balance-factor">2. &nbsp; Node balance factor<a class="headerlink" href="#2-node-balance-factor" title="Permanent link">&para;</a></h3>
<p>The "balance factor" of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as <span class="arithmatex">\(0\)</span>. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p>
<p>The <u>balance factor</u> of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as <span class="arithmatex">\(0\)</span>. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:</p>
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