Balanced Trees

Searching in Balanced and Unbalanced Trees -

• Consider the problem of searching for a node in a tree:
• Balanced trees, of course, give best case behavior


• Unbalanced trees, of course, give worst case behavior


• Some example data that can produce an unbalanced tree, when inserted:
• Sorted
• Reverse sorted
• Alternating large and small keys
• Large sequences of any of the above


• How to avoid this worst case behavior:
• Create a data structure that is guaranteed to be balanced (or close to it)

Balanced Tree Data Structures

• 2-3-4 Trees:
• Have 2, 3, or 4 children per node


• Red-Black Trees: binary tree; each node has a color
• Binary tree
• Each node has is red or black
• A Red node is part its parent (in equivalent 234 tree)
• Maximum depth difference between trees is one half


• AVL Trees: depth of left and right subtrees of each node are kept with one of each other
• height of left and right subtrees of each node are kept within one of each other


• B Trees: many values per node; short trees; used for disk storage
• Many values per node
• Result in short, balanced trees
• Used for disk storage


• B+-Trees
• B tree variant
• Interior nodes are just indices
• All data is in leaves
• Enables easy sequential access