Limiting distributions for additive functionals on Catalan trees

Abstract

Additive tree functionals represent the cost of many divide-and-conquer algorithms. We derive the limiting distribution of the additive functionals induced by toll functions of the form (a) n α when α>0 and (b) log n (the so-called shape functional) on uniformly distributed binary trees, sometimes called Catalan trees. The Gaussian law obtained in the latter case complements the central limit theorem for the shape functional under the random permutation model. Our results give rise to an apparently new family of distributions containing the Airy distribution ( α=1) and the normal distribution [case (b), and case (a) as α↓0]. The main theoretical tools employed are recent results relating asymptotics of the generating functions of sequences to those of their Hadamard product, and the method of moments.