  The problem of characterizing trees with minimal atom-bond-connectivity index
(minimal-ABC trees) has a reputation as one of the most demanding recent open
optimization problems in mathematical chemistry. Here firstly, we give an
affirmative answer to the conjecture, which states that enough large
minimal-ABC trees are comprised solely of a root vertex and so-called $D_z$-
and $D_{z+1}$-branches. Based on the presented theoretical results here and
some already known results, we obtain enough constraints to reduce the search
space and solve the optimization problem, and thus, determine exactly the
minimal-ABC trees of a given arbitrary order.
