EECS 311: Chapter 4 Notes Chris Riesbeck EECS Northwestern

 Unless otherwise noted, all tables, graphs and code from Mark Allen Weiss' Data Structures and Algorithm Analysis in C++, 3 rd ed, copyright © 2006 by Pearson Education, Inc.

Binary Search Trees Binary Search Tree Not a Binary Search Tree

BST with = left/right depths

AVL Trees AVL Tree Not AVL Tree

Right Rotation Rotate right when left left subtree too deep Right subtree of k1… becomes left subtree of k2 Parent pointer to k2… changed to point to k1

Left Rotation Rotate left when right right subtree too deep

Left-Right Rotation Rotate k2 left through k1 first… When right subtree of left subtree too deep …then rotate k2 right through k3

Red-Black Trees  AVL-style rotations to balance  No numbers, just one red-black bit per node (or link)  Use rotations and recoloring to guarantee: Never two red nodes in a row All leaves same depth counting black nodes

Rotations in Red-Black Trees

Recoloring in Red-Black Trees

Splay Tree  Balance on access, not insertion  Use AVL-style rotations  Move item just accessed to the root  Typically makes tree shallower  Complexity Can be O(N) worst case single access O(M log N) on for M accesses Amortized O(log N)  No bookkeeping data needed

Zig-zag Rotation

Zig-zig Rotation

B-Trees  No rotations  N-way tree (N subtrees, N-1 keys)  Keys sorted in each node  When passing through a full node: Make a new node Move the keys and pointers on the right to the new node Move middle key and the pointer to the new node to parent node Note: parent can't be full. Why?

Adding and splitting now add 57

After adding 55 now add 55

After adding 57 now add 40

After adding 40