Recursive tree

Abstract: Consider some model of random finite trees of increasing size. It often happens that the subtree at a uniform random vertex converges in distribution to a limit random tree. We introduce some structure theory for such asymptotic fringe distributions and illustrate with many examples.

Abstract: A process of growing a random recursive tree Tn is studied. The sequence {Tn} is shown to be a sequence of “snapshots” of a Crump–Mode branching process. This connection and a theorem by Kingman are used to show quickly that the height of Tn is asymptotic, with probability one, to c log n. In particular, c = e = 2.718 … for the uniform recursive tree, and c = (2γ)−1, where γe1+γ = 1, for the ordered recursive tree. An analogous reduction provides a short proof of Devroye's limit law for the height of a random m‐ary search tree. We show finally a close connection between another Devroye's result, on the height of a random union‐find tree, and our theorem on the height of the uniform recursive tree. © 1994 John Wiley & Sons, Inc.

Abstract: In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form $X\mathop{=}\limits^{d}\,g((\xi_{i},X_{i}),i\geq 1)$. Here (ξi) and g(⋅) are given and the Xi are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g(⋅) is essentially a “maximum” or “minimum” function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process Xi, are the Xi measurable functions of the innovations process (ξi)?

Abstract: 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 the parameter space of trees is to use global optimization methods like evolutionary algorithms. This paper describes the evtree package, which implements an evolutionary algorithm for learning globally optimal classification and regression trees in R. Computationally intensive tasks are fully computed in C++ while the partykit package is leveraged for representing the resulting trees in R, providing unified infrastructure for summaries, visualizations, and predictions. evtree is compared to the open-source CART implementation rpart, conditional inference trees (ctree), and the open-source C4.5 implementation J48. A benchmark study of predictive accuracy and complexity is carried out in which evtree achieved at least similar and most of the time better results compared to rpart, ctree, and J48. Furthermore, the usefulness of evtree in practice is illustrated in a textbook customer classification task.