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We give a simple graph-theoretical proof that the largest number of maximal independent vertex sets in a tree with n vertices is given by \ m (T) = cases 2^k - 1 + 1& if n = 2k, \\ 2ᵏ & if n = 2k + 1, cases\ a result first proved by Wilf SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 125–130. We also characterize those trees achieving this maximum value. Finally we investigate some related problems.
Bruce E. Sagan (Mon,) studied this question.