Finding the cut vertice of a tree with n vertices that returns connected components with a minimal number of vertices? i need to find an algorithm in O(n) time for my title question for a given tree G = (V,E) with n vertices. It should return the vertice n which after deleting from the tree  results in  a number of connecting components where the maximum number of vertices in a connected component is minimal among all possible cut vertices.
My solution so far is:

*

*s = starting node

*cut_node = s //maybe initialize later in loop

*maxNumComponent = []

if |V| == 0 || |V| == 1
   return null
if |V| == 2 
   return any node
for each v in V with degree[v] > 1 do //maybe do a while loop and mark visited node?
      deleteFromTree(v)
      currentMax = traverse resulting components & return max. number of vertices of all components
      maxNumComponent.add(currentMax)
      if currentMax is min of maxNumComponent
         cut_node=v
      addToTree(v)
return v

I would be happy for any help to optimize this approach or if there is an easier or simpler solution.
Cheers

You could try rooting the tree at any vertex (it doesn't matter), and compute on each node the size of its subtree. This can be done in a single $O(n)$ pass.
Then, using this information, given a node, it should be possible to compute the value you're searching (the $\max$ of the sizes of the remaining components) without new traversals of the tree : for components below in the rooted tree, you have already computed those, and for the (unique) component that includes the root, it has all $n$ vertices except the ones in the subtree, so you know its size.
That whole computation has an (amortized) cost of $O(n)$.
