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Trees possessing no Kekul´e structures (i.e., perfect matching) with the minimal Estrada index are considered. Let Tn be the set of the trees having no perfect matchings with n vertices. When n is odd and n ≥ 5, the trees with the smallest and the second smallest Estrada indices among Tn are obtained. When n is even and n ≥ 6, the tree with the smallest Estrada index in Tn is deduced.