Question

I’ve tried to understand what sorted trees are and binary trees and avl and and and …
I’m still not sure, what makes a sorted tree sorted? And what is the complexity (Big-Oh) between searching in a sorted and searching in an unsorted tree? Hope you can help me.

Answer 1

Binary Trees

There exists two main types of binary trees, balanced and unbalanced. A balanced tree aims to keep the height of the tree (height = the amount of nodes between the root and the furthest child) as even as possible. There are several types of algorithms for balanced trees, the two most famous being AVL- and RedBlack-trees. The complexity for insert/delete/search operations on both AVL and RedBlack trees is O(log n) or better – which is the important part. Other self balancing algorithms are AA-, Splay- and Scapegoat-tree.

Balanced trees gain their property (and name) of being balanced from the fact that after every delete or insert operation on the tree the algorithm introspects the tree to make sure it’s still balanced, if it’s not it will try to fix this (which is done differently with each algorithm) by rotating nodes around in the tree.

Normal (or unbalanced) binary trees do not modify their structure to keep themselves balanced and have the risk of, most often overtime, to become very inefficient (especially if the values are inserted in order). However if performance is of no issue and you mainly want a sorted data structure then they might do. The complexity for insert/delete/search operations on an unbalanced tree range from O(1) (best case – if you want the root) to O(n) (worst-case if you inserted all nodes in order and want the largest node)

There exists another variation which is called a randomized binary tree which uses some kind of randomization to make sure the tree doesn’t become fully unbalanced (which is the same as a linked list)