I need to determine recursion tree depth for strings composed of 10, 100 and 1000 elements when using merge sort. For the 10 elements one/I can do it on a paper, just drawing tree, but what about 100 or 1000 elements? How do I determine recursion tree depth for them? Is there any pattern that I cannot see? What if someone would ask me for recursion tree depth for a string composed of one million elements?
I need to determine the biggest and the smallest recursion tree depth for strings composed of 10, 100 and 1000 when using quicksort. So it's kind of similar but those could be any strings, so is there even constant number that defines the smallest and the biggest? How to count this?
I need to provide a recursive dependency which value for the given $n$ is the number of recursion tree levels of recursive calls of merge sort for an $n$-element string.