CS 411 Fall 2025  >  Outline & Supplemental Notes for October 22, 2025

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CS 411 Fall 2025
Outline & Supplemental Notes
for October 22, 2025

Outline

B-Trees [L 7.4]

• Idea
  • Balanced search trees: way of stored associative data that gives logarithmic time for all CRUD operations.
  • B-Trees: balanced search trees that allow for very large nodes.
  • B-Trees good when we wish to minimize number of nodes accessed. Good for external tables on block-access devices. Make node same size as block?
• Definition
  • Order of a B-Tree: \(m\ge 3\). This is maximum number of children a node can have.
  • Nodes can contain multiple key-value pairs. A non-leaf has one more child than the number of keys it contains.
  • Ordered much like a Binary Search Tree, generalized for nodes containing multiple keys.
  • Root contains \(1\,\dots\,m-1\) keys.
  • Each other node contains \(\lceil m/2 \rceil-1\,\dots\,m-1\) keys.
  • All leaves at same level (so “flat on the bottom”).
• Algorithms
  • Retrieve algorithm
    • Start at root and work down, much like Binary Search Tree.
  • Insert algorithm
    • Find the leaf to insert into (start at root, work down).
    • If space in leaf node, insert there.
    • If leaf would become overfull, then insert new item, split leaf, and insert center item in parent.
    • Work up the tree in this way.
    • If root is overfull, split and make new root.
  • Delete algorithm
    • Will not cover details. Similar to insert, but a bit more complicated. Work down from root, then back up, fixing the tree.
• Analysis
  • CRUD are logarithmic-time, like all balanced search trees.
  • Height of tree is at most something like \(\log_{m/2}n\). Compare Red-Black Tree which has height at most something like \(2\,\log_2 n\). And number of nodes accessed is something like the height of the tree.
• B-Tree vs. B+ Tree
  • We have described a classical B-Tree.
  • Variant: B+ Tree (described in text).
    • Each key in a non-leaf is duplicated in a leaf.
    • Associated values are found only in leaves.
    • So non-leaves function as index into leaves.
    • Leaves may be organized into a linked list.

Supplemental Notes

B-Tree vs. B+ Tree

The B-Tree described in the text is more properly a B+ Tree. In a proper B-Tree, each key occurs in only one node, and the leaves are not organized into a linked list. However, B+ Trees are often called B-Trees, for some reason.