On majorization of closed walks vector of trees with given degree sequences

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 $C(k,T_{\pi }^{\ast })\equiv C(k,T_{\pi }^{\ast },v\in V(T^{\ast }))$, where $T_{\pi }^{\ast }$ is the greedy tree corresponding to $\pi$. In addition, for two trees degree sequences $\pi ,{\pi }^{\mathrm{\prime }}$, if $\pi$ is majorized by ${\pi }^{\mathrm{\prime }}$, then $C(k;T_{\pi }^{\ast })$ is weakly majorized by $C(k;T_{{\pi }^{\mathrm{\prime }}}^{\ast })$.