How do i prove that tree has exactly one center vertex or possibly 2 (bicentral) if the two are neighbour vertexes, i need unformal prove if possible, thanks!
This is certainly no proof. I can give you a method of finding the center (bicenter) of a tree.
Suppose that $\displaystyle n>2$.
First, remove all vertices of degree one along with any incident edges.
We still have a tree remaining. If the number of remaining vertices is more than two, then repeat the process.
If the number of remaining vertices is two you have a bicenter; if it is one you have the center.
i meant when i firstly start, i see all the leafs and get them off the tree altogether with the edges incident with those vertices, now i got new leafs cause previously i deleted the edges that were connecting these new leafs with the vertices that previously were leafs, so now i check if my tree contains two or less vertices if so i stop and i conclude that if there's two vertices one of those could be either center or bicenter vertex (one or another bout cold permute the role), if previously the condition does not hold that means i must continue to remove leafs altogether with the edges incidenting those leafs, eventually ill end up with tree with 1 or 2 vertex. Is that what you talked about?
My question up there was: in the step when you check if in the tree exists more than two vertices, do you check for vertices with degree one, or you check for vertices no matter the degree they got, but it is clear to me now so thanks
as i noted, i always get the root to be one of the vertices to be center/bicenter isn't obvious that i can pick the root and one of it's children as the vertices i look for, cause in a rooted tree if you keep on removing the leafs you will eventually end up with the root and one child to it..
as i told you, the root is the central node always, cause if you just take the tree and redraw it with that all of the child nodes of his you put around him evenly, you will note that the root is actually the center of the graph hence it's central node, and as i told you you can take one of his children randomly and assign it as bicenter node that's it, is just an eye to see it =), thanks anyway!