The Tree of Life and Other Affine Buildings

Andreas Dress, Werner Terhalle

In this note, we discuss some mathematics which has proven to be of use in the analysis of molecular evolution – and, actually, was discovered in this context (cf. [D]).

According to evolutionary theory, the spectrum of present-day species (or biomolecules) arose from their common ancestors according to a well-defined scheme of bi-(or multi-)furcation steps. The task of phylogenetic analysis as defined by E. Haeckel is to unravel that scheme by comparing systematically all data available regarding present and extinct species. This task has been simplified enormously in recent years through the availability of molecular sequence data, first used for that purpose by W. Fitch and E. Margoliash in their landmark paper from 1967 dealing with Cytochrome C sequences [FM]. The basic idea in that field is that species (or molecules) which appear to be closely related should have diverged more recently than species which appear to be less closely related.

A standard formalization is to measure relatedness by a metric defined on the set of species (or molecules) in question. The task then is to construct an (ℝ-)tree which represents the metric (and hence the bifurcation scheme) as closely as possible. Below, we discuss necessary and sufficient conditions for the existence of such a tree that represents the metric exactly, as well as some constructions which lead to that tree if those conditions are fulfilled, and to more or also less treelike structures if not. Remarkably, the theory we developed in this context allowed also to view affine buildings (which in the rank 1 case are ℝ-trees) from a new perspective.

Here are some basic definitions and results:

Definition 1: Given a non-empty set E, an integer m ≥ 2, and a map

v : E^{m} → {−∞} ∪ ℝ,

the pair (E, v) is called a valuated matroid of rank m if the following properties hold:

(VM0) for every e ∈ E, there exist some e_{2}, …, e_{m} ∈ E such that

v(e, e_{2}, …, e_{m}) ≠ −∞,

(VM1) v is totally symmetric,

(VM2) for e_{1}, …, e_{m} ∈ E with #{e_{1}, …, e_{m}} < m, one has

v(e_{1}, …, e_{m}) = −∞,

(VM3) for all e_{1}, …, e_{m}, f_{1}, …, f_{m} ∈ E, one has

v(e_{1}, …, e_{m}) + v(f_{1}, …, f_{m})
≤ max_{1≤i≤m}{v(f_{1}, e_{1}, …, e_{i−1}, e_{i+1}, …, e_{m}) + v(e_{i}, f_{2}, …, f_{m})}.

Condition (VM3) is also called the valuated exchange property.

If {b_{1}, …, b_{m}} ⊆ E satisfies v(b_{1}, …, b_{m}) ≠ −∞, then {b_{1}, …, b_{m}} is called a base of the valuated matroid (E, v).

Note that (VM3) implies the bases exchange property of ordinary matroids for the set B_{(E,v)} of bases of (E, v).

Here is a “generic” example: Let K be a field with a non-archimedean valuation w : K → {−∞} ∪ ℝ, that is a map satisfying the conditions

w(x) = ∞ ⟺ x = 0,
w(x · y) = w(x) + w(y),

and

w(x + y) ≤ max{w(x), w(y)}

for all x, y ∈ K; then – in view of the Grassmann-Plücker identity

det(e_{1}, …, e_{m}) · det(f_{1}, …, f_{m})
= ∑_{i=1}^{m} det(e_{1}, …, e_{i−1}, f_{1}, e_{i+1}, …, e_{m}) · det(e_{i}, f_{2}, …, f_{m})

(e_{1}, …, e_{m}, f_{1}, …, f_{m} ∈ K^{m}) – the pair (K^{m} \ {0}, w ∘ det) is a valuated matroid of rank m.

Definition 2: Given a valuated matroid (E, v) of rank m, we put

T_{(E,v)} := {p : E → ℝ | ∀e ∈ E : p(e) = max_{e_{2},...,e_{m}∈E}{v(e, e_{2}, …, e_{m}) − ∑_{i=2}^{m} p(e_{i})}}.

T_{(E,v)} is also called the tight span of (E, v) or its T -construction.

The following proposition details this set of maps:

Proposition 1: Let H := {(t_{1}, …, t_{m}) ∈ ℝ^{m} | ∑_{i=1}^{m} t_{i} = 0}. Then, for every base

{b_{1}, …, b_{m}} ∈ B_{(E,v)} of a valuated matroid (E, v) of rank m, the map Φ_{b_{1},...,b_{m}} : H → ℝ^{E} which maps each (t_{1}, …, t_{m}) ∈ H to the map

E → ℝ : e ↦ max_{1≤i≤m}\{v(e, b_{1}, …, b_{i−1}, b_{i+1}, …, b_{m}) + t_{i}\} − \frac{m−1}{m}v(b_{1}, …, b_{m})

is an injective map into T_{(E,v)}.

Furthermore, one has

T_{(E,v)} = ⋃_{{b_{1},...,b_{m}}∈B_{(E,v)}} Φ_{b_{1},...,b_{m}}(H).

Thus, T_{(E,v)} is a union of (images of) affine hyperplanes of dimension m−1, called the apartments in T_{(E,v)}.

These apartments intersect as follows:

Proposition 2:

1) Given two bases B, B′ ⊆ E of a valuated matroid (E, v) of rank m, with suitable orderings of their elements as B = {b_{1}, …, b_{m}} and B′ = {b′_{1}, …, b′_{m}}, resp., one has

Φ_{b_{1},...,b_{m}}(H) ∩ Φ_{b′_{1},...,b′_{m}}(H) = ⋂_{i=0}^{m} Φ_{b_{1},...,b_{i},b′_{i+1},...,b′_{m}}(H).

2) Given a base {b_{1}, …, b_{m}} ∈ B_{(E,v)}, an element b_{0} ∈ E \ {b_{1}, …, b_{m}}, and a subset I ⊆ {1, …, m} so that {b_{0}, b_{1}, …, b_{i−1}, b_{i+1}, …, b_{m}} is a base if and only if i ∈ I, then one has

Φ^{−1}(Φ_{b_{1},...,b_{m}}(H) ∩ Φ_{b_{0},b_{1},...,b_{i−1},b_{i+1},...,b_{m}}(H))
= {(t_{1}, …, t_{m}) ∈ H | t_{i} + v(b_{0}, b_{1}, …, b_{i−1}, b_{i+1}, …, b_{m}) =

max_{j∈I}{t_{j} + v(b_{0}, b_{1}, …, b_{j−1}, b_{j+1}, …, b_{m})}}

for every i ∈ I.

We return to our generic example mentioned above, that is, to the valuated matroid (E := K^{m} \ {0}, v = w ∘ det), with K a field with a non-archimedean valuation w : K→ →{−∞} ∪ ℤ. By Γ_{ℤ}, we denote the group of all affine maps from H to itself consisting of a translation by an integer vector and a permutation of coordinates, that is,

Γ_{ℤ} := {γ : H → H |(t_{1}, …, t_{m}) ↦ (t_{σ(1)} + a_{1}, …, t_{σ(m)} + a_{m}) for some
(a_{1}, …, a_{m}) ∈ H ∩ ℤ^{m} and some σ ∈ S_{m}}.

Every subset

C = {Φ_{b_{1},...,b_{m}} ∘ γ(t_{1}, …, t_{m}) | (t_{1}, …, t_{m}) ∈ H with
t_{1} ≤ t_{2} ≤ … ≤ t_{m} ≤ t_{1} + 1}

with {b_{1}, …, b_{m}} ∈ B_{(E,v)} some base and γ ∈ Γ_{ℤ} is called a chamber of T_{(E,v)}; in case {b_{1}, …, b_{m}} is the canonical base of the vector space K^{m} and γ equals id_{H}, the resulting chamber C_{0} is called the fundamental chamber, while the apartment A_{0} = Φ_{b_{1},...,b_{m}}(H) for the canonical base is called the fundamental apartment. If a map p in T_{(E,v)} satisfies p(e) ≡ \frac{i}{m} mod 1 for some i ∈ \{0, …, m − 1\} and every e ∈ E, then p is called a vertex of T_{(E,v)} (of type i). It is easy to see that the general linear group GL_{m}(K) acts transitively on the set of vertices of T_{(E,v)} via its group action defined on T_{(E,v)} by

GL_{m}(K) × T_{(E,v)} → T_{(E,v)} :
(X, p) ↦ (E → ℝ : e ↦ p(X^{−1}e) + \frac{1}{m}w ∘ det(X))

(X ∈ GL_{m}(K), p ∈ T_{(E,v)}, e ∈ E). This action induces a transitive action of the group SL_{m}(K) on the set of apartments as well as on the set of chambers of T_{(E,v)}; since the stabilizers of these actions give rise to a BN-pair in the sense of building theory, one has

Theorem 1: For the valuated matroid (E = K^{m} \ {0}, v = w ∘ det) with K a field with a non-archimedean valuation w : K→ →{−∞} ∪ ℤ, the T -construction T_{(E,v)} is a geometrical realization of the affine building defined for the group GL_{m}(K).

Now, we come back to the general case of an arbitrary valuated matroid (E, v) of rank m.

Lemma 1: For every p ∈ T_{(E,v)}, the map

d_{p} : E × E → ℝ
(e, f) ↦ e^{sup{v(e,f,e_{3},...,e_{m})−p(e)−p(f)− ∑_{i=3}^{m} p(e_{i})|e_{3},...,e_{m}∈E}}

(with e^{−∞} := 0) is a (pseudo-ultra-)metric on E. In addition, for any two maps p and q in T_{(E,v)}, the metrics d_{p} and d_{q} are topologically equivalent.

Definition 3: A valuated matroid (E, v) of rank m is called complete if, for some (or equivalently: for every) p ∈ T_{(E,v)}, the metric space (E, d_{p}) is complete.

Up to “projective equivalence” and identifying “parallel elements” (we refer to [DT1] for details), one has

Theorem 2: Every valuated matroid has an (essentially unique) completion.

(In fact, one only has to complete (E, d_{p}) to a metric space (Ê, d̂) and then to define v̂ : Ê^{m} → {−∞} ∪ ℝ as the continuous extension of v.)

Concerning the T -construction, one has the following result:

Theorem 3: Let (Ê, v̂) be a completion of the valuated matroid (E, v) with Ê ⊇ E. Then the restriction map from T_{(Ê,v̂)} ⊆ ℝÊ to ℝ^{E}, mapping every p ∈ T_{(Ê,v̂)} to p |_{E}, is a bijection into T_{(E,v)}.

From now on, we assume for simplicity (E, v) to be a complete valuated matroid of rank m.

Definition 4: An end of T_{(E,v)} is a map ε from T_{(E,v)} to ℝ satisfying (E1) for every base {b_{1}, …, b_{m}}, there exist some r ∈ {1, …, m}, some affine

map γ : H → H with a coordinate permutation as linear component, and some c ∈ ℝ such that, for every (t_{1}, …, t_{m}) ∈ H, the equation

ε ∘ Φ_{b_{1},...,b_{m}} ∘ γ(t_{1}, …, t_{m}) = max_{1≤i≤r} t_{i} + c

holds;

(E2) there exist some base {b_{1}, …, b_{m}} and some c ∈ ℝ such that, for every

(t_{1}, …, t_{m}) ∈ H,

ε ∘ Φ_{b_{1},...,b_{m}}(t_{1}, …, t_{m}) = t_{1} + c.

The set of all ends of T_{(E,v)} will be denoted by E_{T_{(E,v)}}.

With this definition, one has

Proposition 3: For every e ∈ E, the map

ε_{e} : T_{(E,v)} → ℝ
p ↦ p(e)

is an end of T_{(E,v)}. And, for every ε ∈ E_{T_{(E,v)}}, there exist some e ∈ E and some c ∈ ℝ such that ε = ε_{e} + c.

And one has

Theorem 4: If one defines a map w from the set E_{T_{(E,v)}}^{m} of m-tupels of ends of (E, v) to {−∞} ∪ ℝ by

w(ε_{1}, …, ε_{m}) := inf_{p∈T_{(E,v)}} ∑_{i=1}^{m} ε_{i}(p)

for ε_{1}, …, ε_{m} ∈ E_{T_{(E,v)}}, then one has

w(ε_{e_{1}}, …, ε_{e_{m}}) = v(e_{1}, …, e_{m})

for all e_{1}, …, e_{m} ∈ E. That is, (E_{T_{(E,v)}}, w) is a complete valuated matroid of rank m which – up to “parallel elements” – is isomorphic to (E, v).

We now restrict ourselves to the case that the rank m equals 2. Here, T_{(E,v)} is a path-infinite ℝ-tree, that is an ℝ-tree being the union of isometric images of the real line – namely the apartments from above: for any two p, q ∈ T_{(E,v)}, there exists some base {b_{1}, b_{2}} such that p, q ∈ Φ_{b_{1},b_{2}}(H), say p = Φ_{b_{1},b_{2}}((s, −s)) and q = Φ_{b_{1},b_{2}}((t, −t)) for some s, t ∈ ℝ; then putting d(p, q) := |s − t| leads to a (well-defined) metric on T_{(E,v)} having the desired property. And the ends of T_{(E,v)} in our sense correspond to its ends in the way ends are defined for ℝ-trees, that is, they correspond to (equivalence classes of) isometric embeddings of real halflines into T_{(E,v)}.

An example which we found particularly intriguing is the following one: Let E denote the set of subsets of ℝ which are bounded from above, and for e, f ∈ E, let v(e, f) := sup(e△f) be the supremum of their symmetric difference. Then it is easy to see that (E, v) is a valuated matroid of rank 2. The corresponding ℝ-tree has the particular property that omitting any point leads to the same “number” of connected components, and this number equals #𝔓(ℝ), the cardinality of the powerset of ℝ.

Now, it is well-known that, for the metric d of an ℝ-tree T, the so-called four-point condition

d(x, y) + d(z, w) ≤ max \{ \begin{matrix}d(x, z) + d(y, w), \\ d(x, w) + d(y, z) \end{matrix} \}

holds for all x, y, z, w ∈ T. But this four-point condition is literally the exchange property (VM3) in the rank 2 case! Of course, one has d(x, x) = 0 instead of d(x, x) = −∞ (cf. (VM2)).

This observation led us to the definition of matroidal trees:

Definition 5: A matroidal tree or, for short, matree, is a pair (X, u) consisting of a non-empty set X together with a map u : X ×X → {−∞}∪ℝ satisfying the following three conditions:

(MT0) for every x ∈ X, there exists some y ∈ X with u(x, y) ≠ −∞,

(MT1) u is symmetric,

(MT2) for all x_{1}, x_{2}, y_{1}, y_{2} ∈ X, one has

u(x_{1}, x_{2}) + u(y_{1}, y_{2}) ≤ max \{ \begin{matrix}u(y_{1}, x_{2}) + u(x_{1}, y_{2}), \\ u(y_{1}, x_{1}) + u(x_{2}, y_{2}) \end{matrix} \}

(and no restriction on the diagonal corresponding to (VM2)).

Note that, for every matree (X, u), the restriction u |_{{x∈X|u(x,x)=0}^{2}} is a (pseudo)metric.

Now, let’s have a look at the set

H_{(X,u)} :={f : X → {−∞} ∪ ℝ | f(x) + u(y, z) ≤
max \{ \begin{matrix}f(y) + u(x, z), \\ f(z) + u(x, y) \end{matrix} \} for all x, y, z ∈ X, f ≢ −∞\},

the set of all one-point extensions of a matree (X, u) (containing at least all maps

h_{a} : X → {−∞} ∪ ℝ :
x ↦ u(a, x)

for a ∈ X). If one wants to make a new matree (H_{(X,u)}, w) from this set, and one wants the map w : H_{(X,u)}×H_{(X,u)} → {−∞}∪ℝ to satisfy w(h_{x}, h_{y}) = u(x, y) for all x, y ∈ X (in order to have an “homomorphism” X → H_{(X,u)} : x ↦ h_{x}), and, slightly more general, w(f, h_{x}) = f(x) for every f ∈ H_{(X,u)} and every x ∈ X, then w necessarily has to satify

w(f, g) + u(x, y) ≤ max \{ \begin{matrix}f(x) + g(y), \\ f(y) + g(x) \end{matrix} \}

for all f, g ∈ H_{(X,u)} and all x, y ∈ X. And, indeed, one has

Theorem 5: If, for a matree (X, u) and for H_{(X,u)} as above, one defines

w := w_{(X,u)} : H_{(X,u)} × H_{(X,u)} → {−∞} ∪ ℝ
(f, g) ↦ inf_{x,y∈X} \{max \{ \begin{matrix}f(x) + g(y), \\ f(y) + g(x) \end{matrix} \} − u(x, y)\}

(with the convention (−∞) − (−∞) := +∞), then (H_{(X,u)}, w) is again a matree. In addition, for every f ∈ H_{(X,u)} and every x ∈ X, one has

w(f, h_{x}) = f(x)

– in particular, one has w(h_{x}, h_{y}) = u(x, y) for all x, y ∈ X.

The matree (H_{(X,u)}, w) can be seen as a “hull” of (X, u), as one has

Theorem 6: If, for F ∈ H_{(H_{(X,u)},w_{(X,u)})}, one defines

φ(F) : X → {−∞} ∪ ℝ
x ↦ F (h_{x}),

and, for f ∈ H_{(X,u)},

ψ(f) : H_{(X,u)} → {−∞} ∪ ℝ
g ↦ w_{(X,u)}(f, g),

then φ is a bijective map from H_{(H_{(X,u)},w_{(X,u)})} to H_{(X,u)}, and ψ is a bijective map in the other direction; both maps are inverse to each other; and for all f, g ∈ H_{(X,u)}, one has

w_{(H_{(X,u)},w_{(X,u)})}(ψ(f), ψ(g)) = w(f, g).

Thus, (H_{(H_{(X,u)},w_{(X,u)})}, w_{(H_{(X,u)},w_{(X,u)})}) and (H_{(X,u)}, w_{(X,u)}) are canonically isomorphic matrees.

In addition, one has

Theorem 7: H_{(X,u)} is the smallest set of maps X → {−∞} ∪ ℝ that a) contains {h_{x} | x ∈ X} and b) is closed under addition of constants, under suprema, and under limites. More precisely: for every f ∈ H_{(X,u)}, one of the following three possibilities hold:

(i) there exist some x ∈ X and some c ∈ ℝ such that f = h_{x} + c, (ii) there exist some x, y ∈ X and some b, c ∈ ℝ such that

f = max{h_{x} + d, h_{y} + c},

(iii) there exist sequences (x_{n})_{n∈ℕ} in X and (c_{n})_{n∈ℕ} in ℝ such that

f = lim_{n→∞}(h_{x_{n}} + c_{n}).

Essential for the study of matrees is the following

Fundamental Lemma: Let (X, u) be a matree; for x, y ∈ X with u(x, y) ≠ −∞, put

s_{x,y} := \frac{1}{2}(u(y, y) − u(x, y)) ∈ \{−∞\} ∪ ℝ,

s^{x,y} := \frac{1}{2}(u(x, y) − u(x, x)) ∈ ℝ ∪ \{+∞\},

and I(x, y) := [s_{x,y}, s^{x,y}] ∩ ℝ; for t ∈ ℝ, define h^{t} ∈ H_{(X,u)} by

h^{t} := max\{h_{x} + t, h_{y} − t\} − \frac{1}{2}xy.

Then the map I(x, y) → H_{(X,u)} : t ↦ h^{t} is a surjective isometry onto the set

{f ∈ H_{(X,u)} | w(h_{x}, h_{y}) = w(f, h_{x}) + w(f, h_{y}) and w(f, f) = 0}

– with isometry meaning that w(h^{s}, h^{t}) = |s − t| holds for all s, t ∈ I(x, y).

Corollary: The set

{f ∈ H_{(X,u)} | w(f, f) = 0}

is connected; hence – since the restriction of w to it is a metric satisfying the four-point condition – it is an ℝ-tree relative to the restriction of w (cf. [D]).

We want to close this section by a short discussion on the relationship between H_{(E,v)} and T_{(E,v)} for a valuated matroid (E, v) of rank 2. For this, let

T′_{(E,v)} := {p : E → ℝ | p(e) = sup_{f∈E}{v(e, f) − p(f)} for every e ∈ E}

– note the “sup” instead of “max” as for T_{(E,v)}; and define the canonical metric d on T′_{(E,v)} by d(p, q) := sup_{e∈E} |p(e) − q(e)|. It is easy to see that T′_{(E,v)} is the

completion of T_{(E,v)} relative to this metric. One should remark that T′_{(E,v)} is the set of all minimal elements in the polytope

P_{(E,v)} := {p : E → ℝ | p(e) + p(f) ≥ v(e, f) for all e, f ∈ E}

relative to the order p ≤ q : ⟺ p(e) ≤ q(e) for every e ∈ E.

The Tree of Life and Other Affine Buildings 573

Coming back to the comparison of H_{(E,v)} with T′_{(E,v)}, the following holds: The maps p ∈ T′_{(E,v)} are exactly those maps in H_{(E,v)} satisfying

w_{(E,v)}(p, p) = 0,

and one has

d = w_{(X,u)} |_{{p∈H_{(E,v)}|w_{(E,v)}(p,p)=0}^{2}} .

Slightly more general, one has

{p ∈ H_{(E,v)} | w_{(E,v)}(p, p) ≠ −∞} = {p + c | p ∈ T′_{(E,v)}, c ∈ ℝ}.

And the maps p ∈ H_{(E,v)} satisfying w_{(E,v)}(p, p) = −∞ correspond to the ends of the ℝ-tree T_{(E,v)}.

Based on these considerations, an algorithm for analyzing distance data and for constructing phylogenetic trees if those data fit exactly into trees and phylogenetic networks based on the T-construction if the data do not fit into a tree has been developed jointly with D. Huson and others which is available via

http://bibiserv.techfak.uni-bielefeld.de/splits/ where also further ref-

erences can be found.

References

[BD1] H.-J. Bandelt, A. W. M. Dress: Reconstructing the Shape of a Tree from Observed Dissimilarity Data, Advances in Applied Mathematics 7 (1986), 309-343.

[BD2] H.-J. Bandelt, A. W. M. Dress: A canonical decomposition theory for metrics on a finite set, Advances in Mathematics 92 (1992), 47-105.

[BD3] H.-J. Bandelt, A. W. M. Dress: Split Decomposition: A New and Useful Approach to Phylogenetic Analysis of Distance Data, Molecular Phylogenetics and Evolution 1, No. 3 (1992), 242-252.

[D] A. W. M. Dress: Trees, Tight Extensions of Metric Spaces, and the Cohomological Dimension of Certain Groups: A Note on Combinatorial Properties of Metric Spaces, Advances in Mathematics 53 (1984), 321-402.

[DDH] A. W. M. Dress, J. Dopazo, A. v. Haeseler: Split decomposition: a new technique to analyse viral evolution, PNAS 90 (1993), 10320-10324.

[DEW1] A. W. M. Dress, M. Eigen, R. Winkler-Oswatitsch: Statistical geometry in sequence space: A method of quantitative comparative sequence analysis, Proc. Natl. Acad. Sci. USA 85 (1988), 5913-5917.

[DEW2] A. W. M. Dress, M. Eigen, R. Winkler-Oswatitsch: How old is the genetic code?, Science, 244 (1989), 673-679.

[DMT] A. W. M. Dress, V. L. Moulton, W. F. Terhalle: T-Theory — An Overview, European Journal of Combinatorics 17 (1996) 161-175.

574 Andreas Dress, Werner Terhalle

[DT1] A. W. M. Dress, W. F. Terhalle: A Combinatorial Approach to ℘-adic Geometry, Part I: The Process of Completion, Geometriae Dedicata 46 (1993), 127-148.

[DT2] A. W. M. Dress, W. F. Terhalle: The Real Tree, Advances in Mathematics 120 (1996) 283-301.

[DW] A. W. M. Dress, R. Wetzel: The Human Organism – a Place to Thrive for the Immuno-Deficiency Virus, in: New Approaches in Classification and Data Analysis, ed. E. Diday, Y. Lechevallier, M. Schader, P. Bertrand, B. Burtschy, Springer-Verlag, 1994, 636-643.

[FM] W. M. Fitch, E. Margoliash: The Construction of Phylogenetic Trees — A Generally Applicable Method Utilizing Estimates of the Mutation Distance Obtained from Cytochrome c Sequences, Science 155 (1967), 279-284.

[T1] W. F. Terhalle: Ein kombinatorischer Zugang zu ℘-adischer Geometrie: Bewertete Matroide, Bäume und Gebäude, Dissertation, Universität Bielefeld, 1992.

[T2] W. F. Terhalle: Coordinatizing ℝ-Trees in Terms of Universal c-Trees, Annals of Combinatorics 1 (1997), 183-196.

[T3] W. F. Terhalle: ℝ-Trees and Symmetric Differences of Sets, European Journal of Combinatorics 18 (1997), 825-833.

[T4] W. F. Terhalle: Matroidal Trees. A Unifying Theory of Treelike Spaces and Their Ends, Habilitationsschrift, Universität Bielefeld, submitted.

[WWW1] The www page The Tree of Life

(http://phylogeny.arizona.edu/tree/phylogeny.html) provides uptodate information regarding our present knowledge of (or believe in) the detailed branching structure of species evolution.

[WWW2] The www page Splitstree 2 (http://bibiserv.techfak.uni-bielefeld.de/splits) allows to use or to download the SplitsTree algorithm via the net and to view what that algorithm does to an appropriately specified data input.

[WWW2] The www page Computational Molecular Evolution

(http://dexter.gnets.ncsu.edu/lab/moleevol.html) provides links to (almost) all www pages relevant in this field, from data banks to algorithms (incl. SplitsTree 2) to journals.

Andreas Dress, Werner Terhalle FSP Mathematisierung Universität Bielefeld Postfach 10 01 31 D-33501 Bielefeld dress@mathematik.uni-bielefeld.de terhalle@mathematik.uni-bielefeld.de