The expected value under the Yule model of the squared path-difference distance

The path-difference metric is one of the oldest and most popular distances for the comparison of phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the expected value under the Yule model of evolution of its square on the space of fully resolved rooted phylogenetic trees with n leaves. This complements previous work by Steel-Penny and Mir-Rossell\'o, who computed this mean value for fully resolved unrooted and rooted phylogenetic trees, respectively, under the uniform distribution.Comment: 10 pages, extended version of a paper submitted to Applied Mathematics Lette

Application for medical devices and data management

The purpose of the project is to create an application in Flutter, together with a complete backend ecosystem based on NodeJS and MongoDB that allows the users to connect their medical devices, take measurements, store and access their previous measurements data from a FHIR database in the official medical standardized data format (HL7 FHIR). This data can be used by a manager to remotely keep track of a patients health state and potentially detect anomalies. The objective of this thesis is to provide a base framework for the utilization of daily health-related data generated by simple medical devices for wider uses, such as remote diagnosis, big data studies, etc.OutgoingOutgoingObjectius de Desenvolupament Sostenible::3 - Salut i Benesta

Application for assisted VFR flight planning

The Flutter-based application developed in this study offers pilots a user-friendly interface, intuitive features, and real-time data integration. Through this application, pilots can easily create and modify flight routes, analyze weather patterns, assess airspace restrictions, and carry out simulations to obtain the most efficient route based on multiple parameters, as far as meteorology is concerned. Moreover, the application integrates with various databases and online resources to provide comprehensive data on airports, airspaces, and terrain, enabling pilots to make informed decisions during the flight planning phase. Additionally, this thesis aims to demonstrate that a competent software tool, capable of competing with other established flight planning tools, can be developed by a student. By utilizing the powerful Flutter framework, which provides a rich set of development tools and a vast community support, this research showcases the ability of a student to create a robust and feature-rich application for VFR flight planning. The successful development and evaluation of this application serve as a testament to the potential of aspiring developers to contribute innovative solutions to the aviation industry. Furthermore, this thesis highlights the importance of empowering students and encouraging their involvement in the digital transformation of traditional aviation processes, fostering creativity, and driving progress in the field.OutgoingObjectius de Desenvolupament Sostenible::9 - Indústria, Innovació i InfraestructuraObjectius de Desenvolupament Sostenible::4 - Educació de Qualita

Study of balance and symmetry of rooted trees

The Sacking index of a tree is a measure of balance which has beenintroduced in applications to phylogenetics. The study of this parameterin rooted binary trees, identifying its extremal values and enumerationof trees with given index, has been recently achieved by MareikeFischer. The goal of this project is to extend this analysis to particularly for the classes of rootedk-ary trees for general k. Finally, we will study the symmetry of the binary trees and we will propose some indexes to measure how symmetric is a tre

The mean value of the squared path-difference distance for rooted phylogenetic trees

The path-difference metric is one of the oldest distances for the comparison of fully resolved phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the mean value of the square of the path-difference metric between two fully resolved rooted phylogenetic trees with $n$ leaves, under the uniform distribution. This complements previous work by Steel and Penny, who computed this mean value for fully resolved unrooted phylogenetic trees.Comment: 16 page

Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf

Phylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet. Actually, the cophenetic values of pairs of different taxa alone are not enough to single out phylogenetic trees with weighted arcs or nested taxa. In this paper we define a family of cophenetic metrics that compare phylogenetic trees on a same set of taxa by encoding them by means of their vectors of cophenetic values of pairs of taxa and depths of single taxa, and then computing the $L^p$ norm of the difference of the corresponding vectors. Then, we study, either analytically or numerically, some of their basic properties: neighbors, diameter, distribution, and their rank correlation with each other and with other metrics.Comment: The "authors' cut" of a paper published in BMC Bioinformatics 14:3 (2013). 46 page

The probabilities of trees and cladograms under Ford's α\alpha-model

We give correct explicit formulas for the probabilities of rooted binary trees and cladograms under Ford's $\alpha$-model.Comment: 9 page