We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.

Commonly used classification and regression tree methods like the CART algorithm are recursive partitioning methods that build the model in a forward stepwise search. Although this approach is known to be an efficient heuristic, the results of recursive tree methods are only locally optimal, as splits are chosen to maximize homogeneity at the next step only. An alternative way to search over th...

A recursive probability tree (RPT) is an incipient data structure for representing the distributions in a probabilistic graphical model. RPTs capture most of the types of independencies found in a probability distribution. The explicit representation of these features using RPTs simplifies computations during inference. This paper describes a learning algorithm that builds a RPT from a probabil...

It follows from our results that as n → ∞ the average distance between the root and the (nearer) centroid node of a recursive tree Tn tends to 1; and the average value of the label of the (nearer) centroid node tends to 5/2.

In this paper we present a construction of every k-ary tree using a forest of (k− 1)ary trees satisfying a particular condition. We use this method recursively for the construction of the set of k-ary trees from the set of (k−1)-Dyck paths, thus obtaining a new bijection φ between these two sets. Furthermore, we introduce a new order on [k]∗ which is used for the full description of this biject...

Higher-order recursive schemes (HORS) are schematic representations of functional programs. They generate possibly infinite ranked labelled trees and, in that respect, are known to be equivalent to a restricted fragment of the λY -calculus consisting of ground-type terms whose free variables have types of the form o→ ⋯→ o (with o being a special case). In this paper, we show that any λY -term (...

Higher-order recursive schemes offer an interesting method of approximating program semantics. The semantics of a scheme is an infinite tree labeled with built-in constants. This tree represents the meaning of the program up to the meaning of built-in constants. It is much easier to reason about properties of such trees than properties of interpreted programs. Moreover some interesting properti...

We investigate the problem of checking whether the number of derivation trees of a Datalog program with duplicate semantics is nite or not. We show that given a safe stratiied query and an edb, it is possible to check, in polynomial time, whether the query has a nite number of derivation trees. However, it is undecidable to check whether a safe stratiied query has a nite number of derivation tr...

Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let Tn be the total path length for a random recursive tree of order n. Mahmoud (1991) showed that Wn := (Tn − E[Tn])/n converges almost surely and in L2 to a nondegenerate limiting random variable W . Here we give recurrence relations for the moments of Wn and of W and show that Wn converg...

We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ` selected nodes in a size-n random recursive tree for three different selection rules, namely (i ) isolating all of the nodes labelle...