Definition:Center of Tree

Definition
Take a tree, and remove all the nodes whose degree is $1$, along with all their incident edges.

Repeat the process until either:
 * One node is left

or:
 * Two nodes are left, joined by a single edge.

If one node is left, it is called the center of the tree.

If two nodes are left, joined by a single edge, this is called the bicenter of the tree.

Also see

 * Tree has Center or Bicenter, proving every tree has either a center or a bicenter, but not both.