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An AVL tree is a binary search tree whose child subtree heights differ by at most one; rotations restore balance after updates.
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Formula
h is measured in levels; operations take O(log n) for n stored keys
| Symbol | Meaning | Unit |
|---|---|---|
| Left and right subtree heights | levels | |
| Number of stored keys | count | |
| Balance factor hL−hR | levels |
|h_L-h_R|\le 1The invariant |hL−hR|≤1 forces logarithmic height. Insertion or deletion can violate it locally; single or double rotations restore balance while preserving inorder order. Search, insertion and deletion take O(log n). Height conventions differ on whether an empty subtree has height −1 or 0, but the balance condition is unchanged.
The invariant |hL−hR|≤1 forces logarithmic height. Insertion or deletion can violate it locally; single or double rotations restore balance while preserving inorder order. Search, insertion and deletion take O(log n). Height conventions differ on whether an empty subtree has height −1 or 0, but the balance condition is unchanged.
An AVL tree is a binary search tree whose child subtree heights differ by at most one; rotations restore balance after updates.
Balancefaktor
A valid AVL node has BF −1, 0 or +1.
Use the example for AVL tree balance condition and height to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
An AVL tree is a binary search tree whose child subtree heights differ by at most one; rotations restore balance after updates.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: The invariant |hL−hR|≤1 forces logarithmic height. Insertion or deletion can violate it locally; single or double rotations restore balance while preserving inorder order. Search, insertion and deletion take O(log n)..
Formula cluster
An AVL tree is a binary search tree whose child subtree heights differ by at most one; rotations restore balance after updates.
Insert 30,20,10 into an empty AVL tree. Node 30 becomes left-heavy by two levels. A right rotation makes 20 the root with children 10 and 30. Inorder order stays 10,20,30 and the root’s subtree heights match.
Implement ordered searches with guaranteed logarithmic height.
Here is how to work through a typical AVL tree balance condition and height (Balancefaktor = hL − hR ∈ {−1,0,1}) task step by step:
Task
Use the example for AVL tree balance condition and height to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
An AVL tree is a binary search tree whose child subtree heights differ by at most one; rotations restore balance after updates.
The relation applies only under its stated assumptions.
At every node, left and right subtree heights differ by at most one. With BF=hL−hR, allowed values are −1, 0 and +1. This local rule bounds total height logarithmically, preventing sorted insertions from turning the search tree into a linear chain. An AVL tree preserves binary-search ordering as well as height balance. Rotations must leave the in-order sequence unchanged. The rule is checked at every internal node, rather than only at the root. Choose a consistent empty-subtree height convention; the permitted height difference stays the same.
A rotation changes links among a few nodes while preserving the binary-search in-order sequence. For keys 30,20,10, a right rotation makes 20 parent, with 10 left and 30 right. The sorted order remains 10,20,30 while imbalance is reduced. Implementations must update subtree heights too. An AVL tree preserves binary-search ordering as well as height balance. Rotations must leave the in-order sequence unchanged. Preserving in-order traversal is the main reason a rotation remains a search-tree operation. Subtrees between rotated nodes must remain in their original key-order positions.
A left-right or right-left insertion bends the heavy path inward. Rotate at the child first to create an outer-heavy case, then rotate at the unbalanced node. The case depends on the newest insertion’s direction relative to the heavy child, not only whether the root is left- or right-heavy. An AVL tree preserves binary-search ordering as well as height balance. Rotations must leave the in-order sequence unchanged. The rotation is selected from the pattern of the last two directions on the search path. The two cases are commonly called left-right and right-left rotations.
Since AVL height is O(log n), search visits logarithmically many nodes. Insertions and deletions search a path of that length and perform bounded local rotations per affected node; rebalancing remains logarithmic overall. The guarantee requires balance metadata and more update work than an unbalanced search tree. An AVL tree preserves binary-search ordering as well as height balance. Rotations must leave the in-order sequence unchanged. Other balanced search trees also provide logarithmic bounds but enforce different invariants. A less strictly balanced tree may be preferable under workloads dominated by updates.
Balance depends on subtree height, not numeric distance between keys. Values 30,20,10 illustrate descending insertion, but any x>y>z inserted in that order creates the same left-left imbalance. Rotation changes shape, not keys; search comparisons still put smaller values left of larger values. An AVL tree preserves binary-search ordering as well as height balance. Rotations must leave the in-order sequence unchanged. Equal numerical spacing between node keys is irrelevant to structural tree balance. The numbers illustrate key order only; balance depends on subtree heights.
Quanta exam set
Curated exam set for "AVL tree balance condition and height":
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Which formula describes AVL tree balance condition and height?
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How do you rearrange AVL tree balance condition and height for Balancefaktor?
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Which common mistake happens with AVL tree balance condition and height?
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