  The {\em atom-bond connectivity (ABC) index} is one of the recently most
investigated degree-based molecular structure descriptors, that have
applications in chemistry. For a graph $G$, the ABC index is defined as
$\sum_{uv\in E(G)}\sqrt{\frac{(d(u) +d(v)-2)}{d(u)d(v)}}$, where $d(u)$ is the
degree of vertex $u$ in $G$ and $E(G)$ is the set of edges of $G$. Despite many
attempts in the last few years, it is still an open problem to characterize
trees with minimal $ABC$ index. In this paper, we present an efficient approach
of computing trees with minimal ABC index, by considering the degree sequences
of trees and some known properties of the graphs with minimal $ABC$ index. The
obtained results disprove some existing conjectures end suggest new ones to be
set.
