Combinatorial variations on multidimensional quadtrees

We present a parallel study of three variants of the notion of multidimensional quadtrees: quadtrees of Catalanʹs type, increasing quadtrees, and point quadtrees. Using the language of the theory of species, we present for these models of tree-like structures some combinatorial equations that lead, for example, to several explicit, recursive or asymptotic formulas for the probability of having k nodes in a fixed hyperoctant, the expected value and the variance of the number of leaves.