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We present BeechCOSTe52; a database of European beech (Fagus sylvatica) phenotypic measurements for several traits related to fitness measured in genetic trials planted across Europe. The dataset was compiled and harmonized during the COST-Action E52 (2006–2010), and subsequently cross-validated to ensure consistency of measurement data among trials and provenances. Phenotypic traits (height, diameter at breast height, basal diameter, mortality, phenology of spring bud burst and autumn–leaf discoloration) were recorded in 38 trial sites where 217 provenances covering the entire distribution of European beech were established in two consecutive series (1993/95 and 1996/98). The recorded data refer to 862,095 measurements of the same trees aged from 2 to 15 years old over multiple years. This dataset captures the considerable genetic and phenotypic intra-specific variation present in European beech and should be of interest to researchers from several disciplines including quantitative genetics, ecology, biogeography, macroecology, adaptive management of forests and bioeconomy.

The vertices of thekth power of a directed path withnvertices are exposed one by one to a selector in some random order. At any time the selector can see the graph induced by the vertices that have already appeared. The selector's aim is to choose online the maximal vertex (i.e. the vertex with no outgoing edges). We give upper and lower bounds for the asymptotic behaviour ofpn,kn1/(k+1), wherepn,kis the probability of success under the optimal algorithm. In order to derive the upper bound, we consider a model in which the selector obtains some extra information about the edges that have already appeared. We give the exact asymptotics of the probability of success under the optimal algorithm in this case. In order to derive the lower bound, we analyse a site percolation process on a sequence of thekth powers of a directed path withnvertices.

We examine the evolution of the best choice algorithm and the probability of its success from a directed path to the linear order of the same cardinality through kth powers of a directed path, 1 ≤ k < n. The vertices of a kth power of a directed path of a known length n are exposed one by one to a selector in some random order. At any time the selector can see the graph induced by the vertices that have already come. The selector's aim is to choose online the maximal vertex (i.e. the vertex with no outgoing edges). It is shown that the probability of success pn for the optimal algorithm for the kth power of a directed path satisfies pn = Θ(n −1/(k+1) ). We also consider the case when the selector knows the distance in the underlying path between each two vertices that are joined by an edge in the induced graph. An optimal algorithm for this choice problem is presented. The exact probability of success when using this algorithm is given.

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