Abstract
Classification of graphs with respect to a parameter is an important optimization problem in graph theory. The Merrifield-Simmons index is defined as the total number of independent sets. This index can be calculated by recurrence relation in terms of Fibonacci numbers. In this paper, we characterize the trees whose Merrifield-Simmons indices are calculated by recurrence relation with k steps. Finally, we obtain the extremal values of trees whose Merrifield-Simmons indices are calculated recursively in k steps.
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