Let's consider a tree, rooted at node $1$, where each node has a color (see [CSES Distinct Colors](https://cses.fi/problemset/task/1139)).
For each node, let's store a set containing only that node, and we want to merge the sets in the nodes subtree together such that each node has a set consisting of all colors in the nodes subtree. Doing this allows us to solve a variety of problems, such as query the number of distinct colors in each subtree. Doing this naively, however, yields a runtime complexity of $O(N^2)$.
However, with just a few lines of code, we can significantly speed this up.
When merging two sets, you move from the smaller set to the larger set. If the size of the smaller set is $X$, then the size of the resulting set is at least $2X$. Thus, an element that has been moved $Y$ times will be in a set of size $2^Y$, and since the maximum size of a set is $N$ (the root), each element will be moved at most $O(\log N$) times leading to a total complexity of $O(N\log N)$.
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Additionally, a set doesn't have to be an `std::set`. Many data structures can be merged, such as `std::map` or even adjacency lists.
<info-blocktitle="Challenge">
Prove that if you instead merge sets that have size equal to the depths of the subtrees, then small to large merging does $O(N)$ insert calls.
