Algorithmic counting of nonequivalent compact Huffman codes

It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees (level-greedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively

Characterization of binary trees

In "Level Number Sequences for Trees" Flagotet and Prodinger investigate the problem of counting the number of level number sequences associated to binary trees of $n$ binary nodes. I convert this problem into terms of exterior nodes or "leaves" and leaf number sequences. "Polynomial representation" is then defined to address this problem. Using polynomial representation I find a class of lower bounds to this quantity as well as an exact recursive formula

HPC: Hierarchical phylogeny construction

Rapid improvements in DNA sequencing technology have resulted in long genome sequences for a large number of similar isolates with a wide range of single nucleotide polymorphism (SNP) rates, where some isolates can have thousands of times lower SNP rates than others. Genome sequences of this kind are a challenge to existing methods for construction of phylogenetic trees. We address the issues by developing a hierarchical approach to phylogeny construction. In this method, the construction is performed at multiple levels, where at each level, groups of isolates with similar levels of similarity are identified and their phylogenetic trees are constructed. Time savings are achieved by using a sufficiently large number of columns from the input alignment, instead of all its columns. Our results show that the new approach is 20-60 times more efficient than existing programs and more accurate in situations where highly similar isolates have a wide range of SNP rates

Evaluating the Evolutionary and Genetic Relationships of the Andean Orchids of Northwestern Ecuador

DNA barcoding is a molecular based technique used to separate and identify individual species. Here we establish a DNA Barcode library for the orchid flora of an Andean cloud forest in Northwestern Ecuador. The library contains 135 matK and 136 rbcL DNA Barcodes representing over 33 Orchidaceae genera. Sequence analysis shows percent species resolution was higher for matK (98.8%) than rbcL (70.24%), with a large portion of the unresolved species for the rbcL loci coming from taxonomically complex genera in the subtribe Pleurothallidinae. Neighbor Joining (NJ) trees revealed that the orchid flora of Siempre Verde is divided taxonomically into two large monophyletic clades at the sub family level; Orchidoideae and Epidendroideae. Sequences within Orchidoideae presented with high bootstrap support across all NJ trees (matK, rbcL and matK+rbcL), indicating species within the clade are well resolved. Resolution for sequences within sub family Epidendroideae varied depending on taxonomic clade and loci used. Overall the matK NJ tree outperformed the rbcL NJ tree by delivering monophyletic clades at the subfamily, tribe, and subtribe level with higher bootstrap values, separating a higher number of congeners, particularly those in taxonomically complex genera such as Pleurothallis, Stelis, and Lepanthes. Estimates of evolutionary divergence showed a very low level of intraspecific variation in DNA Barcodes of target cryptic species Oncidium heteranthum, acknowledging that floral traits in Oncidium are often highly plastic, and not indicative of species lines

Molecular phylogeny of brachiopods and phoronids based on nuclear-encoded small subunit ribosomal RNA gene sequences

Brachiopod and phoronid phylogeny is inferred from SSU rDNA sequences of 28 articulate and nine inarticulate brachiopods, three phoronids, two ectoprocts and various outgroups, using gene trees reconstructed by weighted parsimony, distance and maximum likelihood methods. Of these sequences, 33 from brachiopods, two from phoronids and one each from an ectoproct and a priapulan are newly determined. The brachiopod sequences belong to 31 different genera and thus survey about 10% of extant genus-level diversity. Sequences determined in different laboratories and those from closely related taxa agree well, but evidence is presented suggesting that one published phoronid sequence (GenBank accession UO12648) is a brachiopod-phoronid chimaera, and this sequence is excluded from the analyses. The chiton, Acanthopleura, is identified as the phenetically proximal outgroup; other selected outgroups were chosen to allow comparison with recent, non-molecular analyses of brachiopod phylogeny. The different outgroups and methods of phylogenetic reconstruction lead to similar results, with differences mainly in the resolution of weakly supported ancient and recent nodes, including the divergence of inarticulate brachiopod sub-phyla, the position of the rhynchonellids in relation to long- and short-looped articulate brachiopod clades and the relationships of some articulate brachiopod genera and species. Attention is drawn to the problem presented by nodes that are strongly supported by non-molecular evidence but receive only low bootstrap resampling support. Overall, the gene trees agree with morphology-based brachiopod taxonomy, but novel relationships are tentatively suggested for thecideidine and megathyrid brachiopods. Articulate brachiopods are found to be monophyletic in all reconstructions, but monophyly of inarticulate brachiopods and the possible inclusion of phoronids in the inarticulate brachiopod clade are less strongly established. Phoronids are clearly excluded from a sister-group relationship with articulate brachiopods, this proposed relationship being due to the rejected, chimaeric sequence (GenBank UO12648). Lineage relative rate tests show no heterogeneity of evolutionary rate among articulate brachiopod sequences, but indicate that inarticulate brachiopod plus phoronid sequences evolve somewhat more slowly. Both brachiopods and phoronids evolve slowly by comparison with other invertebrates. A number of palaeontologically dated times of earliest appearance are used to make upper and lower estimates of the global rate of brachiopod SSU rDNA evolution, and these estimates are used to infer the likely divergence times of other nodes in the gene tree. There is reasonable agreement between most inferred molecular and palaeontological ages. The estimated rates of SSU rDNA sequence evolution suggest that the last common ancestor of brachiopods, chitons and other protostome invertebrates (Lophotrochozoa and Ecdysozoa) lived deep in Precambrian time. Results of this first DNA-based, taxonomically representative analysis of brachiopod phylogeny are in broad agreement with current morphology-based classification and systematics and are largely consistent with the hypothesis that brachiopod shell ontogeny and morphology are a good guide to phylogeny

Experimental design and statistical rigor in phylogenomics of horizontal and endosymbiotic gene transfer

A growing number of phylogenomic investigations from diverse eukaryotes are examining conflicts among gene trees as evidence of horizontal gene transfer. If multiple foreign genes from the same eukaryotic lineage are found in a given genome, it is increasingly interpreted as concerted gene transfers during a cryptic endosymbiosis in the organism's evolutionary past, also known as "endosymbiotic gene transfer" or EGT. A number of provocative hypotheses of lost or serially replaced endosymbionts have been advanced; to date, however, these inferences largely have been post-hoc interpretations of genomic-wide conflicts among gene trees. With data sets as large and complex as eukaryotic genome sequences, it is critical to examine alternative explanations for intra-genome phylogenetic conflicts, particularly how much conflicting signal is expected from directional biases and statistical noise. The availability of genome-level data both permits and necessitates phylogenomics that test explicit, a priori predictions of horizontal gene transfer, using rigorous statistical methods and clearly defined experimental controls