What does O(log n) mean exactly?

Question

I am learning about Big O Notation running times and amortized times. I understand the notion of O(n) linear time, meaning that the size of the input affects the g

Answer 1

The best way I've always had to mentally visualize an algorithm that runs in O(log n) is as follows:

If you increase the problem size by a multiplicative amount (i.e. multiply its size by 10), the work is only increased by an additive amount.

Applying this to your binary tree question so you have a good application: if you double the number of nodes in a binary tree, the height only increases by 1 (an additive amount). If you double it again, it still only increased by 1. (Obviously I'm assuming it stays balanced and such). That way, instead of doubling your work when the problem size is multiplied, you're only doing very slightly more work. That's why O(log n) algorithms are awesome.