In this work we analyse bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size n, complementing the earlier result of Mahmoud and Smythe (1995, Theoret. Comput. Sci. 144 221–249.) for bucket recursive trees. On the combinatorial side, we define multilabelled generalisations of the tree families d-ary increasing trees and generalised plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection between such bucket increasing tree families and certain families of graphs called increasing diamonds, providing an explanation for phenomena observed by Bodini et al. (2016, Lect. Notes Comput. Sci. 9644 207–219.). Concerning structural properties of bucket increasing trees, we analyse the tree parameter $K_n$. It counts the initial bucket size of the node containing label n in a tree of size n and is closely related to the distribution of node types. Additionally, we analyse the parameters descendants of label j and degree of the bucket containing label j, providing distributional decompositions, complementing and extending earlier results (Kuba and Panholzer (2010), Theoret. Comput. Sci. 411(34–36) 3255–3273.).

This study was performed to investigate the occurrence of livestock-associated methicillin-resistant Staphylococcus aureus (LA-MRSA) in batches of pigs at slaughter and at different stages along the slaughter line. Nasal and ear skin swabs were collected from 105 batches of 10 pigs at six abattoirs. Cultures (pooled or individual) were performed for MRSA using selective media; presumptive MRSA were confirmed by mecA and nuc gene detection and a selection was spa-typed. MRSA was detected in 46 batches. All spa-types detected were those associated with LA-MRSA clonal complex 398. The proportion of positive batches varied among abattoirs (0–100%). Two abattoirs were subsequently further investigated, with samples taken at post-stunning, chiller and either at lairage or post-singe. Results suggested cross-contamination occurred between the lairage and point of post-stunning, but the slaughter processes appeared effective at reducing contamination before carcases entered the chiller. One abattoir provided only negative samples in the initial study and in the subsequent study along the slaughter line (26 batches in total), suggesting differences possibly in the MRSA status of pigs on arrival from supply farms or in its abattoir practices affecting the MRSA status of pigs at the sampling points. This study highlights that in the investigated abattoirs, MRSA was detected in 43.8% of batches of pigs at slaughter using sensitive selective culture methods.

We explore the possibility of combining a knowledge-based reduced order model (ROM) with a reservoir computing approach to learn and predict the dynamics of chaotic systems. The ROM is based on proper orthogonal decomposition (POD) with Galerkin projection to capture the essential dynamics of the chaotic system while the reservoir computing approach used is based on echo state networks (ESNs). Two different hybrid approaches are explored: one where the ESN corrects the modal coefficients of the ROM (hybrid-ESN-A) and one where the ESN uses and corrects the ROM prediction in full state space (hybrid-ESN-B). These approaches are applied on two chaotic systems: the Charney–DeVore system and the Kuramoto–Sivashinsky equation and are compared to the ROM obtained using POD/Galerkin projection and to the data-only approach based uniquely on the ESN. The hybrid-ESN-B approach is seen to provide the best prediction accuracy, outperforming the other hybrid approach, the POD/Galerkin projection ROM, and the data-only ESN, especially when using ESNs with a small number of neurons. In addition, the influence of the accuracy of the ROM on the overall prediction accuracy of the hybrid-ESN-B is assessed rigorously by considering ROMs composed of different numbers of POD modes. Further analysis on how hybrid-ESN-B blends the prediction from the ROM and the ESN to predict the evolution of the system is also provided.