Graphs, partitions and Fibonacci numbers

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number >2^n-1+5 have diameter ?4 and determine the order of these trees with respect to their Fibonacci numbers. Furthermore, it is shown that the average Fibonacci number of a star-like tree (i.e. diameter ?4) is asymptotically for constants A,B as n→∞. This is proved by using a natural correspondence between partitions of integers and star-like trees.