An M-ary search tree allows M-way branching As branching increases, the depth decreases

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Binary Search Trees

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Figure 1983 A 5-ary tree of 31 nodes has only three levels

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The 6-tree is the most popular data structure for diskbound searching

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The B-tree has a host of structure properties

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We can create an M-ary search tree in much the same way we created a binary search tree In a binary search tree, we need one key to decide which of two branches to take In an M-ary search tree, we need M - 1 keys to decide which branch to take To make this scheme efficient in the worst case, we need to ensure that the M-ary search tree is balanced in some way Otherwise, like a binary search tree, it could degenerate into a linked list Actually, we want an even more restrictive balancing condition That is, we do not want an M-ary search tree to degenerate to even a binary search tree because then we would be stuck with log N accesses One way to implement this is to use a B-tree, which is the most popular data structure for disk-bound searching Here, we describe the basic B-tree;3 many variations and improvements exist, and an implementation is somewhat complex because quite a few cases must be addressed However, in principle this technique guarantees only a few disk accesses A B-tree of order M is an M-ary tree with the following proper tie^^

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I The data items are stored at leaves 2 The nonleaf nodes store as many as M - 1 keys to guide the searching; key i represents the smallest key in subtree i + I 3 The root is either a leaf or has between 2 and M children 4 All nonleaf nodes (except the root) have between r ~ 1 2 1 M and

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children 5 All leaves are at the same depth and have between r ~ 1 2 and L data 1 items, for some L (the determination of L is described shortly)

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Nodes must be half full to guarantee that the tree does not degenerate into a simple binary tree

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An example of a B-tree of order 5 is shown in Figure 1984 Note that all nonleaf nodes have between three and five children (and thus between two and four keys); the root could possibly have only two children Here, L = 5 , which means that L and M are the same in this example, but this condition is

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3 What we describe is popularly known as a B+ tree 4 Properties 3 and 5 must be relaxed for the first L insertions (L is a parameter used in property 5)

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Figure 1984 A B-tree of order 5

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not necessary Because L is 5, each leaf has between three and five data items Requiring nodes to be half full guarantees that the B-tree does not degenerate into a simple binary tree Various definitions of B-trees change this structure, mostly in minor ways, but the definition presented here is one of the most commonly used Each node represents a disk block, so we choose M and L on the basis of the size of the items being stored Suppose that one block holds 8192 bytes In our Florida example, each key uses 32 bytes, so in a B-tree of order M, we would have M - 1 keys, for a total of 32M - 32 bytes plus M branches Because each branch is essentially a number of another disk block, we can assume that a branch is 4 bytes Thus the branches use 4M bytes, and the total memory requirement for a nonleaf node is 36M - 32 The largest value of M for which 36M - 32 is no more than 8 192 is 228, so we would choose M = 228 As each data record is 256 bytes, we would be able to fit 32 records in a block Thus we would choose L = 32 Each leaf has between 16 and 32 data records, and each internal node (except the root) branches in at least 114 ways For the 10,000,000 records, there are at most 625,500 leaves Consequently, in the worst case, leaves would be on level 4 In more concrete terms, the worst-case number of accesses is given by approximately l o g , , N, give or take 1 The remaining issue is how to add and remove items from the B-tree In the ideas sketched note that many themes presented earlier recur We begin by examining insertion Suppose that we want to insert 57 into the B-tree shown previously in Figure 1984 A search down the tree reveals that it is not already in the tree We can add it to the leaf as a fifth child, but we may have to reorganize all the data in the leaf to do so However, the cost is negligible compared to that of the disk access, which in this case also includes a disk write

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