Limit laws for embedded trees: Applications to the integrated superBrownian excursion

Abstract

ABSTRACT:We study three families of labeled plane trees. In all these trees, the root is labeled 0 and the labels of two adjacent nodes differ by 0, 1, or −1.One part of the paper is devoted to enumerative results. For each family, and for all j ∈ N, we obtain closed form expressions for the following three generating functions: the generating function of trees having no label larger than j; the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled j; and finally the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled at least j. Strangely enough, all these series turn out to be algebraic, but we have no combinatorial intuition for this algebraicity.The other part of the paper is devoted to deriving limit laws from these enumerative results. In each of our families of trees, we endow the trees of size n with the uniform distribution and study the following random variables: M n , the largest label occurring in a (random) tree; X n ( j), the number of nodes labeled j; and X + n ( j), the number of nodes labeled j or more. We obtain limit laws for scaled versions of these random variables.Finally, we translate the above limit results into statements dealing with the integrated superBrownian excursion. In particular, we describe the law of the supremum of its support (thus recovering some earlier results obtained by Delmas) and the law of its distribution function at a given point. We also conjecture the law of its density (at a given point).

On the degree distribution of the nodes in increasing trees

“…We will not figure out this but refer to [3,18] for applications of that technique to similar types of generating functions. This leads to the following formula for the density f 1 (x) of the limiting distribution of the out-degree of the root:…”

Left and Right Pathlengths in Random Binary Trees

“…In related work, the difference between left and right depths of individual nodes has been studied from a different point of view by Bousquet-Mélou [5]. She considers the generating function of the number of nodes with a given difference and obtains limit results on the distribution of this number.…”

Left and Right Length of Paths in Binary Trees or on a Question of Knuth

“…Here the right (left) depth of a certain node in a binary tree is the number of right (left) edges on the path from the root to the node considered, whereas the right (left) pathlength of a binary tree is the sum of the right (left) depths over all nodes in the tree. In [1], amongst many others, the random variable X n ( j) was studied, which counts the number of nodes in a random binary tree of size n with difference j between the right and the left depth. As one of the main results of [1] the limiting distribution of X n λn 1 4 , λ ≥ 0, was characterized and thus the so-called "vertical profile" of binary trees was well described.…”

“…In [1], amongst many others, the random variable X n ( j) was studied, which counts the number of nodes in a random binary tree of size n with difference j between the right and the left depth. As one of the main results of [1] the limiting distribution of X n λn 1 4 , λ ≥ 0, was characterized and thus the so-called "vertical profile" of binary trees was well described. In [6] all moments E (D r n ) of the difference D n between the right and the left pathlength of random binary trees of size n are computed asymptotically and as a consequence the limiting distribution of the suitably scaled random variable n − 5 4 D n was characterized.…”