The Colless index is one of the most popular and natural balance indices for bifurcating phylogenetic trees, but it makes no sense for multifurcating trees. In this paper we propose a family of Colless-like balance indices C D,f , which depend on a dissimilarity D and a function f : N → R 0 , that generalize the Colless index to multifurcating phylogenetic trees. We provide two functions f such that the most balanced phylogenetic trees according to the corresponding indices C D,f are exactly the fully symmetric ones. Next, for each one of these two functions f and for three popular dissimilarities D (the variance, the standard deviation, and the mean deviation from the median), we determine the range of values of C D,f on the sets of phylogenetic trees with a given number n of leaves. We end the paper by assessing the performance of one of these indices on TreeBASE and using it to show that the trees in this database do not seem to follow either the uniform model for multifurcating trees or the α-γ-model, for any values of α and γ.
Minecraft has become one of the most popular videogames worldwide. Also, their didactical interface, Minecraft Education Edition can be used to generate an educational resource. To provide evidence-based recommendations for the use of this software in multilingual contexts, this chapter presents both some useful tools to propose math problems and an exemplification of the application of this resource for the reinforcement of mathematical content. For this, the theoretical references, the assembly of the server, and the way to implement the practice of mathematics are explained. Also, methodological recommendations of cooperative learning through technological tools with the interest of fostering the reasoning and argumentation in math problems resolutions are given. In conclusion, the inclusion of this game in this educational proposal is expected to increase affectivity towards mathematics.
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