Related papers: Growing Trees and Amoebas' Replications
We present some exact expressions for the number of paths of a given length in a perfect $m$-ary tree. We first count the paths in perfect rooted $m$-ary trees and then use the results to determine the number of paths in perfect unrooted…
Maximum likelihood estimation (MLE) methods are widely used for evolutionary tree. As evolutionary tree is not a smooth parameter, the consistency of its MLE has been a topic of debate. It has been noted without proof that the classical…
We consider branching processes with interaction in continuous time, both with values in the integers and in the reals (in the second case we restrict ourselves to continuous processes), which model the evolution of the size of a…
Various features of the development of individual living species, including individual humans, are programmed. Is death also programmed, and if yes, how is it implemented and what can be the underlying mechanism providing the inevitability…
We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only…
We study a class of evolution models, where the breeding process involves an arbitrary exchangeable process, allowing for mutations to appear. The population size $n$ is fixed, hence after breeding, selection is applied. Individuals are…
A dynamic model for a random network evolving in continuous time is defined where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total…
We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation,…
Infant mortality, by which we understand the postnatal stage during which mortality is declining, is a manifestation and embodiment of congenital abnormalities. Severe defects will translate into death occurring shortly after birth whereas…
A tree $T$ on $2^n$ vertices is called set-sequential if the elements in $V(T)\cup E(T)$ can be labeled with distinct nonzero $(n+1)$-dimensional $01$-vectors such that the vector labeling each edge is the component-wise sum modulo $2$ of…
The mortality rate of many complex multicellular organisms increase with age, which suggests that net aging damage is accumulative, despite remodeling processes. But how exactly do little mishaps in the cellular level accumulate and spread…
We show that the number of $t$-ary trees with path length equal to $p$ is $\exp(h(t^{-1})t\log t \frac{p}{\log p}(1+o(1)))$, where $\entropy(x){=}{-}x\log x {-}(1{-}x)\log (1{-}x)$ is the binary entropy function. Besides its intrinsic…
Aging is a universal consequence of life, yet researchers have identified no universal theme. This manuscript considers aging from the perspective of entropy, wherein things fall apart. We first examine biological information change as a…
We construct an increasing, submultiplicative, arbitrarily rapid function which is not equivalent to the growth function of any finitely generated algebra, demonstrating the difficulty in characterizing growth functions in an asymptotic…
We examine the global organization of growing networks in which a new vertex is attached to already existing ones with a probability depending on their age. We find that the network is infinite- or finite-dimensional depending on whether…
The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary…
Complex systems, in many different scientific sectors, show coarse-grain properties with simple growth laws with respect to fundamental microscopic algorithms. We propose a classification scheme of growth laws which includes human aging,…
A method for creating a forest of model trees to fit samples of a function defined on images is described in several steps: down-sampling the images, determining a tree's hyperplanes, applying convolutions to the hyperplanes to handle small…
Local and global amoebas are families of labeled graphs that satisfy interpolation properties on a fixed vertex set. A labeled graph $G$ on $n$ vertices is a local amoeba (resp. global amoeba) if there exists a sequence of feasible…