Deletion Without Rebalancing in Binary Search Trees

We address the vexing issue of deletions in balanced trees. Rebalancing after a deletion is generally more complicated than
rebalancing after an insertion. Textbooks neglect deletion rebalancing, and many B-tree-based database systems do not do
it. We describe a relaxation of AVL trees in which rebalancing is done after insertions but not after deletions, yet worst-case
access time remains logarithmic in the number of insertions. For any application of balanced trees in which the number of
updates is polynomial in the tree size, our structure offers performance competitive with that of classical balanced trees.With
the addition of periodic rebuilding, the performance of our structure is theoretically superior to that of many if not all classic
balanced tree structures. Our structure needs lg lgm + 1 bits of balance information per node, where m is the number of
insertions and lg is the base-two logarithm, or lg lg n + O(1) with periodic rebuilding, where n is the number of nodes. An
insertion takes up to two rotations and O(1) amortized time, not counting the time to find the insertion position. This is the
same as in standard AVL trees. Using an analysis that relies on an exponential potential function, we show that rebalancing
steps occur with a frequency that is exponentially small in the height of the affected node. Our techniques apply to other
types of balanced trees, notably B-trees, as we show in a companion paper, and in particular red-black trees, which can be
viewed as a special case of B-trees.