Bottom-up Mergesort vs Natural Merge Sort I ran into natural mergesort in this Wikipedia page: https://en.wikipedia.org/wiki/Merge_sort#Natural_merge_sort
However, I can't find much information regarding:

The algorithm(s) to achieve natural mergesort, and the pros and cons of each method
How natural mergesort compares to bottom-up mergesort (pros and cons of each w.r.t the other)

In short, I'm looking for resources that would teach me algorithms to do natural mergesort, and how to weigh it against the usual bottom-up mergesort
Natural Mergesort usually refers to Timsort. In terms of comparison to bottom-up mergesort, Timsort has $O(n)$ best-case running time where as bottom-up mergesort is at least $O(n\log n)$ on any input.
However, analysis of the worst-case running time of Timsort proves to be more tricky; an 2018 paper [1] establishes that its worst-case time complexity is $O(n\log \rho)$, where $\rho$ is the number of maximal monotonic sequences in the input array.
[1]: Nicolas Auger, Vincent Jugé, Cyril Nicaud, Carine Pivoteau, &quot;On the Worst-Case Complexity of TimSort&quot;