A centroid of a tree is a node where removing it splits the tree into components, each with at most nodes. Every tree with nodes has at least one centroid, sometimes two. Think of it as the balance point. No subtree gets too heavy. This property is what makes divide-and-conquer efficient.
When you remove a centroid, you split the problem into smaller, balanced subproblems. For a tree with nodes arranged as a star (one center, four leaves), the center is the centroid. Remove it and you get four components of size each, all well under .