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Abstract.Černý's conjecture asserts the existence of a synchronizing word of length at most (n − 1) 2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p · a r = q · a s for some integers r, s (for a state p and a word w, we denote by p · w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n 2 ). This applies in particular to Huffman codes.

This paper studies the entropy of tree-shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree-shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of the binary Markov tree-shifts over two symbols is either 0 or ln 2. Meanwhile, the realization of a class of reals including multinacci numbers is elaborated , which indicates that tree-shifts are capable of rich phenomena. By considering the influence of three different types of boundary conditions , say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient conditions for the coincidence of entropy with and without boundary conditions are addressed.