B-Tree of degree 3 t=3 data internal nodes 40 80 10 27 31 52 65 88 96 28 30 45 50 70 74 77 82 84 97 99 15 25 33 35 53 55 58 60 62 90 92 94 data

For simplicity, assume each key links to some associated data 40 80 10 27 31 52 65 88 96 3 7 45 50 15 25 82 84 97 99 53 55 58 60 62 28 30 33 35 70 74 77 90 92 94 6/9/2019 IT 279

Bt is a B-tree with minimum degree t (or order m , where m = 2t): The root has n keys with 1  n  2t-1 Each non-root node has n keys with t-1  n  2t-1 Each non-leaf node with n keys has n+1 children All leaves are of the same height For each node as shown in the following k1 k2 ki ki+1 kn where k1 < k2 < ... < ki < ki+1 < ... < kn 6/9/2019 IT 279

Bt is a B-tree with minimum degree t: and if the node is not a leaf, then k1 k2 ki ki+1 kn C0 C1 C2 Ci Ci+1 Cn-1 Cn ki  min(Ci)  max(Ci)  ki+1, for 1  i  n-1 max(C0)  k1 kn  min(Cn) maximum and minimum keys in Ci 6/9/2019 IT 279

Bt is a B-tree with minimum degree t: Some Terminologies ki ki+1 Ci-1 Ci the left-child of key the right-child of key ki ki We say that a node is full if it has 2t-1 keys If t=2, B2 is known as 2-3-4 tree (named after the number of children) 6/9/2019 IT 279

Algorithm for inserting a new key (data) into Bt The new arrived key will be inserted into some leaf Search the leaf with keys that can contain the new key (according to the key values). During the search, if we reach a full node, i.e., it already has 2t-1 keys, then we split the node and repeat 1 and 2 again until the new key is successfully inserted. Note: after splitting a leaf (if necessary), we should have a leaf with room for the new key. 6/9/2019 IT 279

Algorithm for splitting a full root k1 kt-1 kt kt+1 k2t-1 C0 Ct-2 Ct-1 Ct Ct+1 C2t-1 Split  new root kt k1 kt-1 kt+1 k2t-1 C0 Ct-2 Ct-1 Ct Ct+1 C2t-1 6/9/2019 IT 279

Algorithm for splitting a full non-root node I. Splitting node parent node k’ ka ka+1 k’’ link before splitting kt k1 kt-1 kt+1 k2t-1 C0 Ct-2 Ct-1 Ct Ct+1 C2t-1 6/9/2019 IT 279

Algorithm for splitting a full non-root node II. Modifying the parent parent node k’ ka ka+1 k’’ kt k1 kt-1 kt+1 k2t-1 C0 Ct-2 Ct-1 Ct Ct+1 C2t-1 6/9/2019 IT 279

Algorithm for splitting a full non-root node II. Modifying the parent parent node k’ ka kt ka+1 k’’ k1 kt-1 kt+1 k2t-1 C0 Ct-2 Ct-1 Ct Ct+1 C2t-1 6/9/2019 IT 279

t=2, B2 : 2-3-4 tree. The root has n keys with 1  n  2t-1 Each non-root node has n keys with t-1  n  2t-1 t=2, B2 : 2-3-4 tree. 25 10 20 30 40 25 15 16 20 10 30 40 40 40 20 10 20 30 15 16 10 30 20 10 25 30 40 6/9/2019 IT 279

t=2 B2 : 2-3-4 tree. The root has n keys with 1  n  2t-1 Each non-root node has n keys with t-1  n  2t-1 35 t=2 B2 : 2-3-4 tree. 20 10 15 16 25 30 40 35 20 30 20 30 10 15 16 25 35 40 10 15 16 25 40 6/9/2019 IT 279

t=2, B2 : 2-3-4 tree. The root has n keys with 1  n  2t-1 Each non-root node has n keys with t-1  n  2t-1 t=2, B2 : 2-3-4 tree. 5 20 30 10 15 16 25 35 40 5 need to split again 5 20 15 20 30 30 15 10 16 25 35 40 10 16 25 35 40 6/9/2019 IT 279