  Let $d_u$ be the degree of a vertex $u$ of a graph $G$. The atom-bond
sum-connectivity (ABS) index of a graph $G$ is the sum of the numbers
$(1-2(d_v+d_w)^{-1})^{1/2}$ over all edges $vw$ of $G$. This paper gives the
characterization of the graph possessing the minimum ABS index in the class of
all trees of a fixed number of pendent vertices; the star is the unique
extremal graph in the mentioned class of graphs. The problem of determining
graphs possessing the minimum ABS index in the class of all trees with $n$
vertices and $p$ pendent vertices is also addressed; such extremal trees have
the maximum degree $3$ when $n\ge 3p-2\ge7$, and the balanced double star is
the unique such extremal tree for the case $p=n-2$.
