We consider a Bernoulli bond percolation on a random recursive tree of size n ≫ 1, with supercritical parameter pn = 1 − c/ lnn for some c > 0 fixed. It is known that with high probability, there exists then a unique giant cluster of size Gn ∼ e−c, and it follows from a recent result of Schweinsberg [22] that Gn has non-gaussian fluctuations. We provide an explanation of this by analyzing the e...

We present two recursive state space search algorithms that are based on SSS* and Dual*. Both dominate Alpha-Beta on a node count basis and one of them is always faster in searching random trees. These results have been derived on arti cial game trees with negligible leaf node evaluation time. For practical applications with more complex leaf evaluation functions we conjecture that our recursiv...

The parity based divide and conquer recursion trees are introduced where the sizes of the tree do not grow monotonically as n grows. These non-monotonic recursive functions called fogk(n) and f̃ogk(n) are strictly less than linear, o(n) but greater than logarithm, Ω(logn). Properties of fogk(n) such as non-monotonicity, upper and lower bounds, etc. are examined and proven. These functions are us...

We derive several limit results for the profile of random plane-oriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of our proofs are based on a method of moments. We also discover an unexpected connection between t...

Solutions of recursive program schemes over a given signature Σ were characterized by Bruno Courcelle as precisely the context-free (or algebraic) Σ-trees. These are the finite and infinite Σ-trees yielding, via labelling of paths, context-free languages. Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad C “generated” by solutions of recursive ...

The distributions of vertex degrees in random recursive trees and random plane recursive trees are shown to be asymptotically normal. Formulas are given for the asymptotic variances and covariances of the number of vertices with given outdegrees. We also give functional limit theorems for the evolution as vertices are added. The proofs use some old and new results about generalized Pólya urn mo...

Solutions of recursive program schemes over a given signature Σ were characterized by Bruno Courcelle as precisely the context-free (or algebraic) Σ-trees. These are the finite and infinite Σ-trees yielding, via labelling of paths, context-free languages. Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad C “generated” by solutions of recursive ...

Solutions of recursive program schemes over a given signature Σ were characterized by Bruno Courcelle as precisely the context-free (or algebraic) Σ-trees. These are the finite and infinite Σ-trees yielding, via labelling of paths, context-free languages. Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad CH “generated” by solutions of recursive...

We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n?2 new un is added to its parent chosen randomly according some rule. In weighted tree, we choose uk among {u1,u2,…,un?1} with probability proportional wk, where (wn)n?1 deterministic sequence that fix beforehand. affine preferential attachment fitnesses, choosing an...