Extremal trees with respect to Hosoya Index and Merrifield-Simmons Index

We characterize the trees T with n vertices whose Hosoya index (total number of matchings) is Z(T ) > 16fn−5 resp. the trees whose Merrifield-Simmons index (total number of independent subsets) is σ(T ) < 18fn−5 + 21fn−6, where fk is the kth Fibonacci number. It turns out that all the trees satisfying the inequality are tripodes (trees with exactly three leaves) and the path in both cases. Furthermore, we show that the remarkable correspondence Z(T )+σ(T ) = fn+3 holds for all these trees. These results are achieved by modifying and enhancing methods due to Li and Zhao, who found the trees of secondand third-smallest Merrifield-Simmons index.