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# Lect8 - B-Trees 1 B-trees Why Best tree discussed so far...

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1 B-Trees

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2 B-trees: Why? Best tree discussed so far – AVL Tree: Important operation Findkey () can be implemented in O(log n) time. AVL Tree has problems for large data the size of the AVL tree increases and may not fit in the system’s main memory. the height of the AVL tree also increases – Findkey() operation no more efficient.
3 B-trees: Why? To overcome these problems, m-way trees have been created. M-way tree allows: Each node to have at the most m children (or sub-trees) Each non-leaf node has (k-1 ) keys if it has k children. M-way tree is ordered and could be balanced like an AVL tree

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4 An m-way Tree E K P B D G R S T M N Z X Y M-way tree of order 4
5 B-trees: Why? Because in a m-way tree, a node can have more than two children and more than one data element in it, the overall size (i.e. number of nodes) decreases height decreases. Also, at any time only a part of the tree can be loaded into the main memory – the rest of the tree can remain in disk storage. B-trees are a kind of m-way trees. B+-trees are special types of B-trees. Database files are represented as B-trees.

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6 B+ Tree : Properties B+ Tree of order M has following properties: Root is either a leaf or has 2 to M children. Non-leaf nodes (except the root) have M/2 to M children which means they can have from M/2 -1 to M-1 keys stored in them. All leaves are at the same depth or level Data elements are stored in the leaves only and have between M/2 and M data elements.
7 B+ Tree : Properties Non-leaf nodes store at the most M-1 keys to guide search; key i represents the smallest key in the subtree i + 1. Note : 1. Actually leaf nodes can have up to L data elements. To simplify we assume L is equal to M. 2. Choice of parameters L and M depends on the data being stored in the B+-tree.

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8 B+ Tree : Example 1 21 72 48 12 15 59 25 41 31 84 91 1,4,8,11 91,92,99 12,13 15,18,19 48,49,50 21,24 25,26 31,38 41,43,46 59,68 72,78 84,88 B+ Tree of Order 4 i.e. M=4
9 B+ Tree : Example 2 22: _ 16: _ 41: 58 8, 11, 12 16, 17 22, 23, 31 41, 52 58, 59, 61 B+ Tree of Order 3 This is how we shall draw a B+ Tree: non-leaf nodes as ovals and leaf nodes as rectangles.

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10 B+ Tree : Search How is FindKey operation performed in a B+- Tree? Almost as in a BST– the keys in the non-leaf
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Lect8 - B-Trees 1 B-trees Why Best tree discussed so far...

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