On a tree-cutting problem of P. Ash (Q1182874)

It is shown that in every tree with \(N\) vertices, there are \(k\) vertices such that the connected components obtained by deleting those vertices can be partitioned into two classes \(C^ k_ 1\), \(C^ k_ 2\) with \[ \delta_ k(T)=\| C^ k_ 1\|-\| C^ k_ 2\|<\max\left(1,3^{-k}\left(N-{3^ k-1\over 2}\right)\right). \] Moreover, for each \(k\) there is an infinite family of trees with the optimal value \(\delta_ k(T)=3^{-k}(N-{3^ k-1\over 2})-2(k-1)\). The corresponding questions for the edge deletion are also considered.