  The {\em atom-bond connectivity (ABC) index} is a degree-based molecular
descriptor, that found chemical applications. It is well known that among all
connected graphs, the graphs with minimal ABC index are trees. A complete
characterization of trees with minimal $ABC$ index is still an open problem. In
this paper, we present new structural properties of trees with minimal ABC
index. Our main results reveal that trees with minimal ABC index do not contain
so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more
than four $B_4$-branches.
