the *undirected path* in $ T $ between the vertices $ v_{i} $ and $ v_{j} $ is denoted by $ v_{i}Tv_{j}:=v_{i}e_{i (i+1)}v_{i+1}e_{(i+1) (i+2)}\dots v_{j-1}e_{(j-1) j}v_{j} $. Similarly, $\overrightarrow{v_{i}Tv_{j}}:=v_{i}\overrightarrow{e_{i(i+1)}}v_{i+1}\overrightarrow{e_{(i+1) (i+2)}}\dots\break v_{j-1}\overrightarrow{e_{(j-1) j}}v_{j}  $ denote the *directed path* in $ T $ from the vertex $ v_{i} $ to the vertex $ v_{j}$. A *rooted tree* is a tree in which one vertex has been fixed as the root. Let $T$ be a rooted tree with the vertex $v_{r}$ as the root. The vertex set of the rooted tree $T$ admits a canonical partial ordering on it as follows: $ v_{x} \leq v_{y}$ if the vertex $ v_{x} $ lies in the path $ v_{r}Tv_{y} $. This partial order is called the *tree-order* on $ V(T) $ associated with the rooted tree $ T $ with root vertex $ v_{r} $ [@Diestel]. A rooted spanning tree $ T $ of a connected graph $ G $ is said to be a *normal spanning tree* if any two adjacent vertices of $G $ are comparable with respect to the tree ordering. Whenever we consider $ T $ as a normal spanning tree of a connected graph $ G $, we assume that 