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
British English spells this centre.

The convention on is to use the American English spelling center, but it is appreciated that there may be lapses.