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 $π$, then for any tree with degree sequence $π$, the sequence $C(k;T)\equiv(C_{v}(k;T), v\in V(T))$ is weakly majorized by the sequence $C(k, T_π^*)\equiv C(k, T_π^*, v\in V(T^*))$, where $T_π^*$ is the greedy tree corresponding to $π$. In addition, for two trees degree sequences $π,~π'$, if $π$ is majorized by $π'$, then $C(k;T_π^*)$ is weakly majorized by $C(k;T_{π'}^*)$.