The set of all permutations with n symbols is a symmetric group denoted by Sn. A transposition tree, T , is a spanning tree over its n vertices VT =1, 2, 3, . . . n where the vertices are the positions of a permutation π and π is in Sn. T is the operation and the edge set ET denotes the corresponding generator set. The goal is to sort a given permutation π with T . The number of generators of ET that suffices to sort any π ∈ Sn constitutes an upper bound. It is an upper bound, on the diameter of the corresponding Cayley graph Γ i.e. diam(Γ ). A precise upper bound equals diam(Γ ). Such bounds are known only for a few tress. Jerrum showed that computing diam(Γ ) is intractable in general if the number of generators is two or more whereas T has n − 1 generators. For several operations computing a tight upper bound is of theoretical interest. Such bounds have applications in evolutionary biology to compute the evolutionary relatedness of species and parallel/distributed computing for latency estimation. The earliest algorithm computed an upper bound f (Γ ) in a Ω(n!) time by examining all π in Sn. Subsequently, polynomial time algorithms were designed to compute upper bounds or their estimates. We design an upper bound δ * whose cumulative value for all trees of a given size n is shown to be the tightest for n ≤ 15. We show that δ * is tightest known upper bound for full binary trees. 4
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