  Let $C_{v}(k;T)$ be the number of the closed walks of length $k$ starting at
vertex $v$ in a tree $T$. We prove that for a given tree degree sequence $\pi$,
then for any tree with degree sequence $\pi$, the sequence
  $C(k;T)\equiv(C_{v}(k;T), v\in V(T))$ is weakly majorized by the sequence
$C(k, T_{\pi}^*)\equiv C(k, T_{\pi}^*, v\in V(T^*))$, where $T_{\pi}^*$ is the
greedy tree corresponding to $\pi$. In addition, for two trees degree sequences
$\pi,~\pi'$, if $\pi$ is majorized by $\pi'$, then $C(k;T_{\pi}^*)$ is weakly
majorized by $C(k;T_{\pi'}^*)$.

Combinatorics