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2620001 Data Structures

This article explores B+-Trees and Standard Tries, highlighting their structures and functionalities. B+-Trees are balanced trees designed for efficient searching, allowing between [n/2] and n children per node while maintaining equal path lengths from root to leaf. The article details operations such as searching (log_d(n)), insertion, and deletion. For Standard Tries, a character-labelled ordered tree is described, showcasing how strings are derived from its structure. Additionally, Threaded Binary Trees are discussed, addressing space efficiency in inorder traversals and their respective traversal algorithms.

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2620001 Data Structures

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  1. 2620001Data Structures Search Trees

  2. B+ - Tree Structure • A B+ - Tree is in the form of a balanced tree in which every path from the root of the tree to a leaf of the tree is the same length. • Each nonleaf node in the tree has between [n/2] and n children, where n is fixed. • B+ - Trees are good for searches, but cause some overhead issues in wasted space.

  3. B+ Trees: Summary • Searching: • logd(n) – Where d is the order, and n is the number of entries • Insertion: • Find the leaf to insert into • If full, split the node, and adjust index accordingly • Similar cost as searching • Deletion • Find the leaf node • Delete • May not remain half-full; must adjust the index accordingly

  4. Standard Tries • The standard trie for a set of strings S is an ordered tree such that: • Each node but the root is labeled with a character • The children of a node are alphabetically ordered • The paths from the external nodes to the root yield the strings of S • Example: standard trie for the set of strings S = { bear, bell, bid, bull, buy, sell, stock, stop }

  5. Trie / Suffix Tree • A tree representing a set of strings. c a { aeef ad bbfe bbfg c } b e b d e f c f e g

  6. Threaded Binary Trees • Binary trees have a lot of wasted space: the leaf nodes each have 2 null pointers • We can use these pointers to help us in inorder traversals • We have the pointers reference the next node in an inorder traversal; called threads • We need to know if a pointer is an actual link or a thread, so we keep a boolean for each pointer

  7. Threaded Tree Code • Example code: class Node { Node left, right; boolean leftThread, rightThread; }

  8. Threaded Tree Example 6 8 3 1 5 7 11 9 13

  9. Threaded Binary Tree Traversal • We start at the leftmost node in the tree, print it, and follow its right thread • If we follow a thread to the right, we output the node and continue to its right • If we follow a link to the right, we go to the leftmost node, print it, and continue

  10. Threaded Tree Traversal Output 1 3 6 8 3 1 5 7 11 9 13 Follow thread to right, print node

  11. Threaded Tree Traversal Output 1 3 5 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print

  12. Threaded Tree Traversal Output 1 3 5 6 6 8 3 1 5 7 11 9 13 Follow thread to right, print node

  13. Threaded Tree Traversal Output 1 3 5 6 7 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print

  14. Threaded Tree Traversal Output 1 3 5 6 7 8 6 8 3 1 5 7 11 9 13 Follow thread to right, print node

  15. Threaded Tree Traversal Output 1 3 5 6 7 8 9 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print

  16. Threaded Tree Traversal Output 1 3 5 6 7 8 9 11 6 8 3 1 5 7 11 9 13 Follow thread to right, print node

  17. Threaded Tree Traversal Output 1 3 5 6 7 8 9 11 13 6 8 3 1 5 7 11 9 13 Follow link to right, go to leftmost node and print

  18. Example - 2

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