Indexed Dendrograms on random Dissimilarities

Isomorphism and Symmetries in Random Phylogenetic Trees

“…In the latter case, arguments that entirely parallel those applied to unlabeled trees give us that z → F (z, u/2) has no singularity on |z| = 1 2 . This implies, for u ∈ , the exponential smallness of ϕ n (u/2), as defined in (18), resulting in an estimate that parallels (21). Theorem IX.14 of [8] again enables us to conclude as to the existence of a local limit law.…”

“…As expressed by Theorem IX.14 of [8], the existence of a quasi-powers approximation (when u is near 1), as in (16) and (17), and of the exponentially small bound (when u ∈ is away from 1), as provided by (21), suffices to ensure the existence of a local limit law. (The reasoning corresponding to Theorem IX.14 of [8] is simple: start from…”

“…Use (21) to neglect the contribution corresponding to u ∈ ; appeal to the saddle point method applied to the quasi-powers approximation to estimate the central part u ∈ , and conclude.) (ii) The labeled case (Y n , B n ).…”

“…From a statistician's point of view, it may be of interest to determine the probability for two trees to be similar (rather than plainly isomorphic), given some structural similarity distance between nonplane trees-see, for instance, [21] for a study conducted under probabilistic assumptions that differ from ours. Combinatorial generating functions can still be useful in this broad range of problems, as we now show by considering the following: determine the probability that two randomly chosen trees, τ and τ , of the same size have the same number of symmetrical nodes.…”

Morphological segmentation of yeast by image analysis

“…A region that remains isolated for a long time may be significantly different from the other. With respect to the ID's, the discrete survival time of a singleton r (original catchment basins) [15] is the integer l(l) defined by r 2P lK1 and r ;P l (3)…”