Index

age (of a species), 218 clique, 3 distance function, 113 agreement forest, 31 cluster, 18 diversification rate r ,212 Aldous β-splitting model, 55 cluster network, 253 diversity function, 145 algebraic set/variety, 192 coalescent point process (CPP), diversity indices, 138 antichain, 258 217 double-cut-and-join (DCJ) Apresjan cluster, 114 coalescent time units, 225 operation, 121 automorphism, 7 Colless index, 53 double-start tree, 206 compatible set of trees, 21 Baker–Campbell–Hausdorff compressed network, 251 edge (BCH) formula, 157 conditioned critical branching interior, 4 balanced minimum evolution process, 220 pendant, 4 (BME), 126 consensus subdividing, 4 Bayesian star paradox, 183 Adams, 38 equal input model, 158 biconnected component, 247 frequency-difference, 37 equal splits (ES) index, 139 Billera–Holmes–Vogtmann loose, 38 equivalent phylogenetic (BHV) space, 131 majority (+), 37 networks, 238 block strict, 21 equivalent phylogenetic trees, 9 of a network, 247 consensus function evolutionary distance, 150 of a partition, 3 anonymity of a, 39 evolutionary distinctiveness budgeted nature reserve problem neutrality of a, 39 (ED), 139 (BNRP), 138 consistency (of distance based excess of set of trees, 70 BUILD algorithm, 68 tree reconstruction), 121 excess-free set of trees, 75 Buneman graph, 26 continuous realization, 150 exchangeability property (EP), convex partition, 88 52 capture, 92 covarion model, 151 expansion (of a network), 253 Catalan number, 28 cover digraph, 19, 252 caterpillar, 11 cyclic permutation, 27 fair proportion (FP) index, 139 Cayley’s formula, 15 cyclomatic number, 5 Fano’s lemma, 178 center, 8 Fibonacci sequence, 96, 104, 194 central symmetry edge, 8 decisive sets, 82 field of bullets model, 143 central symmetry vertex, 8 deep coalescent cost, 231 simple, 143 centroid, 7 define(atree),73 Fitch–Hartigan algorithm, 100 character, 87 dense set of rooted triples, 263 forest, 4 binary, 87 determining set ,127 four-point condition, 112 partial binary, 98 digraph, 6 strongly compatible, 89 disentangling trees, 85 Galton–Watson branching trivial, 89 display (a tree), 64 process, 199 cherry, 4 a set of trees, 65 gamma statistic, 220 circular distance function, 243 by a directed network, 254 gene tree, 224 circular ordering, 28 by an undirected network, 239 anomalous, 227 circular split system, 27 display graph, 71 general Markov model, 154

291 292 Index general time-reversible (GTR) least common ancestor (LCA) parsimony operation, 100 model, 156 mapping, 230 parsimony score, 100 graph, 3 Linnean classification, 18 partition intersection graph, 90 bipartite, 3 lowest stable ancestor, 263 partition of sets, 3 chordal, 5 patchwork, 76 diameter of, 3 majority rule, 187 path, 3 directed, 6 ancestral estimate, 104 pattern, 165 directed acyclic (DAG), 6 consensus tree, 20 perfect phylogeny, 88 k-partite, 3 Markovian property, 52 perfect phylogeny haplotyping Gromov–Farris transform, 117 matching, 259 (PPH), 99 group elimination, 52 matrix representation with persistent perfect phylogeny parsimony (MRP), 109 (PPP), 98 Hadamard matrix, 167 maximal agreement set (MAS), Petersen graph, 33, 132 Hall’s theorem, 259 79 phylogenetic Hamming distance, 119 maximal agreements subtree ideal, 193 hardwired cluster, 256 (MAST), 79 invariant, 192 Hellinger distance, 177 maximum agreement forest mixture on T ,171 Helly property, 4 (MAF), 31 network, 238 hierarchy, 18 maximum compatibility, 100 terrace, 83 Hilbert’s basis theorem, 192 maximum compatible tree, 80 variety, 193 Hilbert’s Nullstellensatz, 192 maximum likelihood (ML) phylogenetic X -tree homoplasy score, 102 estimation, 147 binary, 10 homoplasy-free, 87 tree, 180 restricted, 63 homotopy, 132, 190 maximum minimum distance rooted, 9 hybridization, 232 (MMD), 135 unrooted, 10 hybridization network, 252 maximum parsimony (MP) tree, phylogenetic diversity (PD), 133 hyperbolicity, 132 107 index, 138 median hull, 244 over Abelian groups, 144 identifiability condition, 179 median vertex, 5 unrooted, 133 identify (a phylogeny), 94 Menger’s theorem, 101, 105 phylogenetic forest incompatible clusters, 36 minimal extension, 100 unrooted binary, 18 incomplete directed perfect minimum evolution phylogenetics, 1 phylogeny haplotyping character-based, 100 phylogeny, 12 (IDPP), 99 model invariant, 193 pigeonhole principle, 20 incomplete lineage sorting (ILS), molecular clock, 157 product partition, 38 224 multispecies coalescent model, profile of phylogenies, 36 irreducible Markov process, 151 225 pyramid, 130 isometric function, 240 multivariate Lagrange inversion, indexed, 130 107 Jacobi’s identity, 150 mutual information, 178 quartet metric, 80 Jukes–Cantor (JC69) model, 156 Möbius function, 191, 198 quartet tree, 12, 65 Möbius inversion, 198 quaternary relation, 27 Kalmanson metric, 129 Kimura’s three-substitution nearest neighbor interchange random cluster model, 161 (K3ST) model, 156 (NNI), 29 ranked tree Kimura’s two-substitution neighbor joining (NJ), 121 oriented, 46 (K2ST) model, 156 Neighbor-Net, 242 unlabeled binary, 48 Noah’s Ark problem (NAP), 144 ranking function, 45 labeled history, 45 nondegenerate (set of trees), 66 redundant arc, 251 lateral gene transfer (LGT), 224, refinement 232 orbit-stabilizer theorem, 41 of a partition, 3 network, 260 of a tree, 21 leaf of a network, 238 Pólya’s urn, 44, 58 restriction (of a tree), 63 leaf-centroid, 8 pairwise compatibility, 23 reticulate arc, 246 least common ancestor (LCA), 6 parallel edges, 241 reticulation vertex, 245 Index 293 reticulation-visible network, 249 strict consensus MRP supertree, tree-child network, 247 Riccati equation, 212 109 tree-sibling network, 248 Robinson–Foulds (RF) distance, subdominant ultrametric, 115 treewidth, 6 20, 25 submodular function, 137 trinet, 263 rooted subtree and prune (rSPR), substitution probability, 164 triplet cover, 128 32 subtree prune and regraft (SPR), trivial clusters, 19 rooted triple, 65 30 trivial invariant, 194 suppressing vertices of degree 2, Tuffley poset, 190 Sackin index, 53 4 safety radius symbolic ultrametric, 117 ultrametric, 113 edge, 124 symmetry group, 7 ultrametric edge lengths, 114 l ,l ,124 ∞ 2 uniform model, 50 stochastic, 124 tanglegram, 43 uniform Poisson model for LGT, sampling consistency, 52 temporal 234 Sauer–Shelah lemma, 23 labeling, 250 unlabeled ranked tree (URT) Shapely value index, 140 network, 250 distribution, 219 Shapley value, 141 ternary relation, 22 shellability, 131, 191 topology invariant, 193 valid extension (F ), 87 singe-start tree, 206 trace of a matrix, 150 vertex representation of an site-to-site rate variation, 173 transition matrix, 148 ultrametric, 114 slim set of trees, 70 tree, 4 softwired cluster, 256 binary, 6 span, 65 perfect, 11 weak hierarchy, 23 species tree, 224 refinement of a, 21, 24 Wedderburn–Etherington split rooted, 6 number, 42 trivial, 23 star, 4 Wiener index, 31 X -, 23 tree arc, 246 split graph, 240 tree bisection and reconnection X -forest, 191 split network, 241 (TBR), 30 X -split, 23 star tree, 4 tree containment problem X -tree, 12 stationary (TCP), 255 distribution, 149 tree metric, 111 Yule–Harding (YH) distribution model, 151 tree representation, 111 model, 43 Stirling’s approximation, 16 tree vertex, 245 unrooted, 57