Which tree has the smallest ABCABC index among trees with kk leaves?

Given a graph GG, the atom–bond connectivity (ABCABC) index is defined to be ABC(G)=∑u∼vd(u)+d(v)−2d(u)d(v) where uu and vv are vertices of GG, d(u)d(u) denotes the degree of the vertex uu, and u∼vu∼v indicates that uu and vv are adjacent. Although it is known that among trees of a given order nn, the star has maximum ABCABC index, we show that if k≤18k≤18, then the star of order k+1k+1 has minimum ABCABC index among trees with kk leaves. If k≥19k≥19, then the balanced double star of order k+2k+2 has the smallest ABCABC index.