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AVL Search trees: Height-balanced binary trees

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• Height-balanced tree
  • AVL tree:

    The AVL tree was invented by G.M. Adelson-Velsky and E.M. Landis Property of the AVL tree: For every node v of the AVL tree: | height(left child) - height(right child) | ≤ 1 (If a node does not have a left child, then height(left child) = 0) (If a node does not have a right child, then height(right child) = 0)

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  • Example:

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  • Counter example:

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• AVL tree is a recursive structure
  • Notice that the definition of the AVL tree says every node:

    For every node v of the AVL tree: | height(left child) - height(right child) | ≤ 1

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  • This means:

    Every subtree of an AVL tree is also an AVL tree !!!!

    Example:

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• Advantages and disadvantages of AVL trees
  • Advantage:

    Better search times for keys (As we will see, the running time for a findKey(k) operation in a AVL tree is guaranteed to be O(log(n))

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  • Disadvantage:

    Longer running times for the insert and remove operations (As we will see, the insert and remove operation must re-balance the AVL tree....)