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