1 The number of independent sets in a treeA set of vertices in a graph G is called independent if the set induces no edges. Wewrite i(G) for the number of independent sets in G; i(G) is often known as the Fibonaccinumber, or in mathematical chemistry as the Merrifield-Simmons index or the σ-index.The study was initiated by Prodinger and Tichy in . In particular, they showed thatamong trees of the same order, the maximum and minimum Fibonacci numbers are at-tained by the star and the path respectively. The name stems from the fact that theFibonacci numbers of paths are the usual Fibonacci numbers. Indeed, as the empty setis independent, i(P0) = 1, i(P1) = 2 and i(Pn) = i(Pn− 1) + i(Pn− 2) for n > 2.The inverse question asks for a positive integer k, whether there exists a graph G suchthat i(G) = k. Clearly there does as i(Kk− 1) = k (note that the empty set is independent).The...