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.

Such a tree is called a central tree.

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

Such a tree is called a bicentral tree.

Also see
From Tree has Center or Bicenter, every tree is either central or bicentral, but not both.

Note
The UK English spelling of this is centre.