Data Structures and Algorithm Analysis - Computer Science at ...

354 Chap. 10 Indexing18332320233020301921 24 31 19 21 24 31(a)(b)2318331220304810 15 19 21 24(c)3145 47 50 52Figure 10.14 Example of inserting a record that causes the 2-3 tree root to split.(a) The value 19 is added to the 2-3 tree of Figure 10.9. This causes the nodecontaining 20 and 21 to split, promoting 20. (b) This in turn causes the internalnode containing 23 and 30 to split, promoting 23. (c) Finally, the root node splits,promoting 23 to become the left record in the new root. The result is that the treebecomes one level higher.merged together. The delete operation for the 2-3 tree is excessively complex andwill not be described further. Instead, a complete discussion of deletion will bepostponed until the next section, where it can be generalized for a particular variantof the B-tree.The 2-3 tree insert and delete routines do not add new nodes at the bottom ofthe tree. Instead they cause leaf nodes to split or merge, possibly causing a rippleeffect moving up the tree to the root. If necessary the root will split, causing a newroot node to be created and making the tree one level deeper. On deletion, if thelast two children of the root merge, then the root node is removed and the tree willlose a level. In either case, all leaf nodes are always at the same level. When allleaf nodes are at the same level, we say that a tree is height balanced. Because the2-3 tree is height balanced, and every internal node has at least two children, weknow that the maximum depth of the tree is log n. Thus, all 2-3 tree insert, find,and delete operations require Θ(log n) time.