We consider Boolean functions over n variables. Any such function can be represented (and computed) by a complete binary tree with and or or in the internal nodes and a literal in the external nodes, and many different trees can represent the same function, so that a fundamental question is related to the so-called complexity of a Boolean function: L(f) := minimal size of a tree computing f.The existence of a limiting probability distribution P (·) on the set of and/or trees was shown by Lefmann and Savický [8]. We give here an alternative proof, which leads to effective computation in simple cases. We also consider the relationship between the probability P (f) and the complexity L(f) of a Boolean function f. A detailed analysis of the functions enumerating some sub-families of trees, and of their radius of convergence, allows us to improve on the upper bound of P (f), established by Lefmann and Savický.
Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree ' may be described by the number of nodes or the number of leaves in layer t n , respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic functions obtained by means of generating functions arising from the combinatorial setup and complex contour integration. Besides, an integral representation for the two-dimensional density of Brownian excursion local time is derived.
Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree may be described by the number of nodes or the number of leaves in layer t √ n, respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic functions which are obtained by means of generating functions arising from the combinatorial setup and complex contour integration. Besides, an integral representation for the two dimensional density of Brownian excursion local time is derived.
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