Hi, I want to share a very interesting proof for the existence of centroid.
Let's direct each edge to the side that has more vertices. (direct randomly if equal) Then you have n vertices and n-1 edges, therefore at least one vertex has an outgoing degree of 0. (If for every vertex the outgoing degree was at least one, we needed at least n edges) This vertex is your centroid. (Because all edges are directed towards it, so no child has more than n/2 vertices).
Using this approach, not only you prove centroid exists, but you can also find it.
I explained it in detail and with visuals in my video on centroid #2 here: https://www.youtube.com/watch?v=t74JJlMIQv8
Recently many people wanted me to talk more about more advanced topics. So I thought maybe I should share this here. (not many people had watched the video)
If you like such facts, let me know. I will make more videos about them.
Let me know about your opinions.