  The modified Albertson index, denoted by $A\!^*\!$, of a graph $G$ is defined
as $A\!^*\!(G)=\sum_{uv\in E(G)} |(d_{u})^{2}- (d_{v})^{2}|$, where $d_u$,
$d_v$ denote the degrees of the vertices $u$, $v$, respectively, of $G$ and
$E(G)$ is the edge set of $G$. In this note, a sharp lower bound of $A\!^*$ in
terms of the maximum degree for the case of trees is derived. The $n$-vertex
trees having maximal and minimal $A\!^*$ values are also characterized here.
Moreover, it is shown that $A\!^*\!(G)$ is non-negative even integer for every
graph $G$ and that there exist infinitely many connected graphs whose $A\!^*$
value is $2t$ for every integer $t\in\{0,3,4,5\}\cup\{8,9,10,\cdots\}$.
