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B⁺-tree: introduction

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• B⁺-tree
  • B⁺-tree:

    B+-tree = an acronym for a tree structure that nobody knows what the acronym stands for.... Possible meaning: Balanced tree (because the tree is balanced Boeing tree (because reportedly, someone working for Boeing developed it)

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  • What is a B⁺-tree:

    B+-tree = a dynamic multi-level index Organized as a balanced tree structure

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• Definitions related to a tree in general

  • Definitions:

    Root node = the node at the "top" (or bottom depending how you draw the tree) of the tree Internal node = a node that has one or more child node(s) Leaf node = a node that does not have any children nodes

    Note:

    B+-tree is a height balanced tree: Every leaf node is located at the same distance (height) from the root node

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• Storing a B⁺-tree
  • How to store a B⁺-tree index:

    Each (internal/leaf) node is stored as one disk block

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• Structure of an internal node of a B⁺-tree
  • Structure of an internal node (including the root node):

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  • Properties of an internal node:

    Each internal node is stored in one block on disk A internal node contains at most: ≤   n       keys ≤   n+1   pointers (= database addresses) to other B+-tree nodes A full internal node: A pointer in an internal node points to: A (another) B+-tree node (= a disk block !!!) Every node (except the root node) has ≥   ⌈ (n+1)/2 ⌉ pointers A minimal internal node:

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  • Summary: (and a concrete example with n = 3)

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• Structure of a leaf node of a B⁺-tree
  • Structure of an leaf node :

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  • Properties of an leaf node:

    Each leaf node is stored in one block on disk A leaf node contains at most: ≤   n   keys ≤   n   pointers (= database addresses) to data blocks (= database addresses) The last pointer points to the next leaf node (= a disk block !) in the B+-tree A full leaf node: Every leaf node (except when the leaf node is the root node) has ≥   ⌈ n/2 ⌉ keys A minimal leaf node:

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    Summary: (and a concrete example with n = 3)

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• Difference between a B-tree and a B⁺ tree
  • B-tree:

    B-tree: The leaf nodes in a B-tree is not linked I.e.: The last record pointer in a leaf node is not used

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• In this course
  • We will only use B⁺-trees because:

    It's very little effort to maintain the links in the leaf nodes We need the ability to traverse the records in an ordered manner The links between the leaf nodes allows us to traverse the records in an ordered manner !!!!

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  • Therefore:

    Whenever I mentioned: B-tree           I really mean: B+-tree !!!

    (There are a few places in the notes where I forgot to put the super script)

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• Summary: structural requirements of a B⁺-tree
  • The following figure summarizes the structural requirement of a B⁺-tree:

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  • Content of an internal node:

    k search key values k+1 pointers to other B+-tree nodes

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  • Content of an leaf node:

    k search key values k pointers to data blocks 1 pointer to the next leaf (sibbling) node

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• Properties of the search key values in an internal node of a B⁺-tree node
  • Search keys stored in an internal node statisfies the following property:

    Explanation:

    All search keys in the left-most (= first) subtree is smaller (<) the value k1 All search keys in the 2nd subtree have values:   k1 ≤ key value < k2 All search keys in the 3rd subtree have values:   k2 ≤ key value < k3 And so on....

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  • Example:

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• Search keys and block pointers in a leaf node
  • Search keys and block/record pointer stored in an leaf node statisfies the following property:

    Note that

    A block pointer = a database address of a block

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• Computing n:   the number of search keys (and pointers) in a node of the B⁺-tree
  • Fact:

    A node (both internal and external) of the B+-tree is stored in a disk block

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  • Consequently:

    We will always pack a disk block with the maximum number of search keys !!!!

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  • Computing n (the number of search keys in a node):

    Let: Sk = length of a search key (in bytes) Sp = length of a pointer (in bytes) In 1 node, there are maximum: n search keys = n × Sk bytes n+1 pointers = (n+1) × Sp bytes Therefore: We must find the largest value of n such that n × Sk + (n+1) × Sp ≤ BlockSize

    Example:

    BlockSize = 4096 bytes Sk = 4 bytes (search key) Sp = 8 bytes Then: 4n + 8(n+1) ≤ 4096 12n + 8 ≤ 4096 12n ≤ 4088 n ≤ 340.666 ===> n = 340 search keys in 1 block

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• Example of a B⁺-tree
  • Example B⁺-tree (with n = 3 , #pointers = 4):

    Structure requirement:

    Internal node: 2   ≤   # pointers   ≤   4        Leaf node: 2   ≤   # keys   ≤   3