Annals of Combinatorics Springer-Verlag 1999

Transcription

1 Annals of Combinatorics 3 (1999) 1-12 Annals of Combinatorics Springer-Verlag 1999 Patching Up X-Trees Sebastian B6cker 1., Andreas W.M. Dress lt, and Mike A. Steel 25 I GK Strukturbildungsprozesse, FSP Mathematisierung, Universit~it Bielefeld, PF , Bielefeld, Germany {boecker, 2Biomathematics Research Centre, University of Canterbury, Private Bag 4800, Christchurch, New Zealand m. math. canterbury, ac. nz Received January 8, 1999 AMS Subject Classification: 05C05, 92D15 Abstract. A fundamental problem in many areas of classification, and particularly in biology, is the reconstruction of a leaf-labeled tree from just a subset of its induced subtrees. Without loss of generality, we may assume that these induced subtrees all have precisely four leaves. Of particular interest is determining whether a collection of quartet subtrees uniquely defines a parent tree. Here, we solve this problem in the case where the collection of quartet trees is of minimal size, by studying encodings of binary trees by such quartet trees. We obtain a characterization of minimal encodings that exploits an underlying "patchwork" structure. As we will show elsewhere, this allows one to obtain a polynomial time algorithm for certain instances of the problem of reconstructing trees from subtrees. Keywords: trees, supertrees, tree amalgamation, quartet encodings of trees, hierarchies, patchworks 1. Introduction Trees are widely used to represent evolutionary, historical, or hierarchical relationships in various fields of classification. In biology for example, such trees ("phylogenies") typically represent the evolutionary history of a collection of extant species or the line of descent of some gene [ 17]. They may also be used to classify individuals (or populations) of the same species [6]. In historical linguistics, trees have been used to represent the evolution of languages [18], while in the branch of philology known as stemmatology, trees may represent the way in which different versions of a manuscript arose through successive copying [ 12]. * Supported by "DFG - Graduiertenkolleg Stmkturbildungsprozesse", Forschungsschwerpunkt Mathematisierung, University of Bielefeld, Germany. t Currently visiting professor at The City College, The City University of New York, Department of Chemical Engineering, Convent Avenue at 140th Street, New York, NY 10031, USA. * Supported by the New Zealand Marsden Fund.

2 2 S. B6cker, A.W.M. Dress, and M.A. Steel In most of these applications, the objects of interest occur at the tips (leaves) of the tree, and all other vertices of the tree correspond to a branching (or speciation) event. From a mathematical perspective, we have a finite set X 1 of objects of interest (species, languages, etc.), and we consider triples T = (V, E; ~) consisting of a tree (V, E) with finite vertex set V = VT and edge set E = ET C_ (v), and a map ~ = CT "X ~ V such that ve~(x) holds for all vev with degt(v)_<2, (1.1) where degt(v ) denotes the degree #{e E ET : v E e} of the vertex v C VT. A triple T = (V, E; ~) that satisfies these conditions is henceforth called an X-tree. Denoting the set of vertices of degree i by Vi, we define an X-tree T = (V, E; t~) to be a phylogenetic X-tree if ~ is a bijection from X onto the set I/1 of leaves of (V, E). In addition, if every vertex in V is of degree either 1 or 3 (in which case (V, E) is called a binary tree), then we will say that T = (V, E; ~) is a binary X-tree. Two X-trees T = (V, E; ~) and T' = (W, U; ~') are isomorphic if there exists a bijection ct : V ~ ~ V' that induces a bijection between E and E t and satisfies ~ = cto~ (in which case there is exactly one such map tx). As is well known, there is a canonical and useful one-to-one correspondence between (isomorphism classes of) X-trees and certain set systems due to P. Buneman (cf. [5]) that we shall now recall. A split of X is a subset {A, B} C_ P(X) such that A, B is a bipartition of X into two non-empty, disjoint subsets; apartial split of X is a split of some non-empty subset of X. We let S(X) (resp. Span(X)) denote the set of all splits ofx (resp. all partial splits of X). Two splits $1, $2 are called compatible if there exist A 1 E $1 and A2 E $2 with A 1 fq A2 = 0. A partial split {A, B} is trivial if min{#a,#b} = 1. A partial split $1 = {A1, B1} is said to extend a partial split $2 = {A2, B2 } if $2 = { (A2 U Be) NA1, (A2 U B2) N B1 } (that is, A2 C A1 and B2 C_ B1, or A2 C B1 and B2 C_ Al) holds in which case we will also write Sz < Sx. Now, each edge e in an X-tree T = (V, E; ~) gives rise to a split of X-- simply delete e from E and apply ~-1 to the two connected components of the resulting graph (V, E - {e}) to obtain a split that we will call a T-split and denote by S[e] = ST[e]. Let S[T] denote the set of all T-splits. It is easily checked that distinct edges induce distinct splits and that the set S[T] is compatible, that is, any two splits from S[T] are compatible. Furthermore, we have #5[T] < 2#X - 3 with equality precisely if T is a binary X-tree. This follows easily from the following fundamental property of trees: If, for a tree (V, E), we denote the set of inner edges not incident with a leaf by/~ C E, then #8 <_ #vl +#v2-3 (1.2) holds for every finite tree (V, E), while equality holds for a tree with #V2 = 0 if and only if that tree is binary. Buneman established the following fundamental correspondences in [5]: Lemma 1.1. The map T ~-~ S[T] induces bijections between (i) the set of(isomorphism classes of) X-trees and the set of compatible split systems s c_ s(x) l Throughout this paper, let #X denote the cardinality of a finite set X, and P(X) the set of its subsets.

3 Patching Up X-Trees 3 (ii) the set of(isomorphism classes of) binary X-trees and the set of compatible split systems S C_ S(X) for which #S = 2#X - 3 holds, or equivalently, the set of maximal compatible split systems. This correspondence between X-trees and compatible split systems provides a convenient partial order on the set of (isomorphism classes of) X-trees: We write T I < T precisely if S[T'] C_ S[T]. Now, given an X-tree T = (V, E; ~) and a non-empty subset Y C_ X, we obtain an induced Y-tree Tit as follows: First, construct the minimal subtree (V', E') of (V, E) that connects all vertices from ~(Y). Then make this tree "homeomorphically irreducible" by replacing each maximal path running (except for its two end points) through degreetwo vertices from W - ~(Y) only, by a single edge (and deleting the superfluous vertices and edges) to obtain a tree (Vy, Er). The restriction #It =: ~" maps Y into Vy and satisfies condition (1.1) (with X, V, and T replaced by Y, Vr, and Tit, respectively). We call the resulting Y-tree Tit = (Vv, Ev; Cv) the induced (Y-)subtree of T. We will say that an X-tree T displays a Y-tree T v if T ~ <_ TIv holds. A more succinct but less visual description of TIv is, in view of Lemma 1.1, to describe its set of splits: S[TIv ] = {S' E 5(Y) : S' <_ S holds for some S E S[T]} ={{ANY, BnY}: {A,B}ES[T] and AnY, BNY~O}. (1.3) Our interest lies in the reverse reconstruction problem: Given as input a collection of subtrees, we wish to determine whether we can amalgamate them into a common supertree, that is, whether there exists some X-tree that displays these subtrees, and if so, whether there exists exactly one such tree. Formally, let (Y1,..., Yk) be a family of non-empty subsets of X, and consider a family if" := (T1,..., Tk) where Tj is a Yj-tree for j = 1,...,k. We may wish to consider the set 'T(ff) of all (isomorphism classes of) phylogenetic X-trees T that display every tree in F. As we will see in the following section, the related and seemingly more special task of reconstructing trees from partial splits is actually equivalent to the task described above; so, the problem of computing T(F) can always be reduced to this particular version of the reconstruction problem. There are several reasons why such reconstruction problems arise naturally in applications such as biology. Firstly, we may wish to combine trees that have been reconstructed using distinct, though overlapping, collections of species (usually by different researchers using different data and, as often as not, different reconstruction methods). A second reason is that, in general, it is difficult to accurately reconstruct large trees directly, and we may choose instead to reconstruct trees for small subsets and then combine these in a parent tree (or parent trees) (see [1,10,11,16,19]). A third reason is that, for genetic data, the number of sites that can be accurately aligned across a small number of closely related sequences is generally much larger than the corresponding number of sites for a set that is large and includes rather diverse sequences. If we were to construct if" by estimating a tree for every subset of size 4, then (as #q'(f) < 1 must hold, cf. [1]) we can easily compute 'T(F) and will usually find that T(ff-) = 0 holds, that is, some of the subtrees must have been incorrectly estimated (this was already known to Colonius and Schulze [7, 8], see also [1, 19]). Thus, we may wish to use only those subtrees that are strongly supported by the data (usually

4 4 S. B6cker, A.W.M. Dress, and M.A. Steel involving closely related objects), and so we will generally have available trees for only a small number of subsets of X. This makes the reconstruction problem more difficult computationally but, of course, also more gratifying whenever one is led this way to a simultaneously non-empty and well-supported set of trees. Of course, we could examine all X-trees to determine which (if any) of them display (every tree in) F; however, this is computationally infeasible since even the number of non-isomorphic binary X-trees grows super-exponentially with the number of leaves. Indeed, this number is precisely the product (2#X - 5) of the first (#X - 2) odd numbers, a result that dates back to 1870 (see [ 13]). This motivates the results described below. We wish to thank David Torney for some encouraging and helpful comments on this work. 2. Partial Splits Given a partial split S = {A, B} E Spa~ (X), we define the support of S by S := A U B, and for every x E _S, we define S(x) := a ifx E A and S(x) := B otherwise; for,5 C_ 5part(X), we define,5 := UsesS. For natural numbers i, j, let ~,j(x) := {{A,B} E Spart(X) : {#A,#B} = {i, j}}. (2.1) A partial split Q E Q(X) := 52,2(x) is called a quartet split, and we use Q = xylwz as shorthand for a = {{x, y}, {w, z}}. Given a subset S C_ Spart(X) of partial splits of X and an X-tree T, we say that T is concordant with S if, for every {A, B} E S, there exists at least one edge e of T that separates (a) from ~(B), that is, with {A, B} < S[e]. Let Tx(S) = T(`5) denote the set of all (isomorphism classes of) phylogenetic X-trees concordant with,5. Note that T E T(S) and T < T t implies T' E T(S), hence, any X-tree T with {T} = T(`5) for some set S C Spart(X) must be a binary X-tree. Note also that a collection S of partial splits with T(,5) = {T} for some binary X-tree T = (V, E; ) must contain at least one partial split for every inner edge that specifically "fits" this edge and, hence, S must contain at least #/~ = #X- 3 distinct nontrivial splits in view of inequality (1.2). It is easily shown [1, 14] that, for S c_ `spart(x) and the relation Q(S) := {Q E Q(X) : Q < S for some S E S}, (2.2) T(S) = T(Q(S)) must hold. Thus, there is no loss of generality in restricting one's attention to quartet splits when reconstructing (phylogenetic) trees from partial splits. Similarly, we have = for any family F = (Tl,..., Tk) as above, so the problem of reconstructing trees from subtrees also reduces to the problem of reconstructing them from (quartet) splits.

5 Patching Up X-Trees 5 3. Quartet Encodings In this section, we analyze conditions under which a binary X-tree is uniquely determined by selecting, for each inner edge e, a corresponding representative quartet split Q with Q <S[e]. In order to describe our main result (Theorem 3.11), we must introduce more terminology and preliminary results. Definition 3.1. Let T = (V, E; d?) denote a binary X-tree. A map q : E -+ Q(X) is called a quartet encoding oft if S[e] extends q(e) for each e E E. For every e E and F C E, we define q_(e):=q(e)cx, q(f):=(q(e): eef}cq(x), and q_(f):=uq(e). eef We will say that q defines T if T is the only phylogenetic X-tree concordant with q(e,). A quartet encoding q is tight if, for each edge e E E, there exists no other edge in separating the two subsets in q(e). It is easy to see (cf. [14]) that a quartet encoding that defines a tree T is tight; furthermore, given a binary X-tree T and a tight quartet encoding q of T, then q(/~) = X and #q(e) = #X - 3 holds. It is also easy to see that T(Q) = {T} holds for some set of quartet splits Q with #Q = #X - 3 and some (necessarily binary) X-tree T if and only if there exists a quartet encoding q of T with Q = q(e) that defines T. We now present three instructive examples of tight encodings. Example 3.2. For X := {1,...,6}, consider the binary X-tree T1 = (V, E; d~l := Idx) with E := {el,..., eg} having the nontrivial splits and S[el] = {{1, 2}, (3, 4, 5, 6}}, S[e2] = {{3, 4), (1,2, 5, 6}}, S[e3] = {(5, 6}, {1, 2, 3, 4}}, plus the six trivial splits as depicted in Figure l(a). Consider the quartet encoding q : E = {el, ez, e3} -4 Q(X) defined by q(el) := 12145, q(e2) := and q(e3) := (3.1) Then q is a tight encoding of Ta, but does not define T1, as it also encodes the X-tree T2 depicted in Fig. 1 (b). A construction generalizing this example can be used to prove the non-existence of consensus methods that are equivariant and Pareto on subtrees, see [15]. Example 3.3. Let X := {1,...,7}, and suppose T = (V, E; d~ := Idx) is a caterpillar with seven leaves as depicted in Figure 2, and q :E --~ Q({1,...,7)) is the following quartet encoding: q(el) := 12136, q(e2) := 13146, q(e3) := 24157, and q(e4) := Then it is easy to check that q defines T.

6 S. B6cker, A.W.M. Dress, and M.A. Steel (a) (b) Figure 1: Two possible binary trees for Example 3.2. el e2 e3 e4 Figure 2: Caterpillar with seven leaves. Example 3.4. Suppose q is any tight quartet encoding of a binary X-tree with ["] q(e) 0. (3.2) e6~ Then it has been observed in [14] -- and it follows immediately from Theorem 3.11 below -- that q defines T. To exploit inequality (1.2), we now introduce the following definition: Definition 3.5. Given a binary X-tree T = (V, E; OO), a subset F C E, and a quartet encoding q : E --+ Q(X), the (q-)excess off is given by We say F is (q-)excess-free /f exc(f) = 0 holds. exc(f) = excq(f) := #q(f) - #F - 3. (3.3) Note that exc({e}) = 0 holds for every e E/~, and that exc(0) = -3. Lemma 3.6. Suppose q is a tight quartet encoding of a binary X-tree T = (V, E; d~). Then (i) exc(k~) = 0; (ii) for every non-empty subset F C ~, one has exc(f) > 0; (iii) if F is excess-free, then F is a connected subset of edges in T, and F equals the set of inner edges ofrlqip

7 Patching Up X-Trees 7 Sketch of proof O) This is merely restating that a binary tree with n leaves has n - 3 inner edges, cf. [5] for example. (ii) Consider T]y = (Vy, Ey; ~r) with Y := _q(f) and note that, as q is tight, we have F C_ (Er). Hence, (1.2) implies #F _< #(ey) _< #r- 3 = #_q(f) - 3, as claimed. (iii) As above, we consider Tit = (Vr, Er; ~r) for Y := q(f). If F were not connected, then F C (ET) since (Ey) is connected. From (1.2), we would conclude #E < #(Ey) < #Y - 3 and, hence, exc(f) > 0. To illustrate the usefulness of these concepts, we now show how they provide further constructions of encodings that define a binary X-tree T. First, we make a further definition: Let us say that two distinct elements x, y of X form a pair of twins of T if the two edges incident with ~(x) and ~(y), respectively, share a vertex v = v(x,y), in which case the third edge incident with v is denoted by e(x, y). In Example 3.2, for instance, and ~ form a pair of twins for the tree depicted in Figure l(a) with e(~, ~) = el. If #X > 4, every binary X-tree has at least two pairs of twins, and one has e(x, y) E for every pair x, y of twins. Example Suppose T = (V, E; ) is a binary X-tree such that the inner edges of T are labeled E = {el,..., en-3}, and that q is a tight quartet encoding of T with Then q defines T. / H \ #(q(ei)\[,.jq(ej)}=l for i=2,...,n-3. (3.4) j<i Proof We apply induction on n. The result holds for n = 4, so suppose it holds for 4,...,n - 1 and that #X = n. Let T ~ denote another binary X-tree that is concordant with q(e). By assumption, there exists x E X with x E q(en-3), but x q(ej) for j = 1,...,n-4. We define Y :=X- {x} and F := {el,...,en-4}, then qlf defines TIt as well as T'Ir and, hence, Tit ~ T'Ir by the induction hypothesis. It remains to show that x is attached to the same edge of T and T'. To this end, we infer from Lemma 3.6 (iii) that F is connected, since q is tight and exc(f) = 0. So x must have a twin in T, denoted y E Y, and {x, y} E q(en-3) must hold. But the same holds true for T' which indeed implies T --- T ~. Example 3.8. For a binary X-tree T = (V, E; ~), define clus(t) := t,)eees[e], the set of clusters of T. Suppose f : clus(t) --+ X is any function that satisfies the condition f(a) EA forall A E clus(t). (3.5) Then f defines a tight quartet encoding of T, denoted qf, as follows: For each inner edge e = {vl, v2}, deletion of Vl and v2 and their incident edges partition T into four connected components and, thereby, it partitions X into four sets {A1, A2, B1, Be},

8 8 S. B6cker, A.W.M. Dress, and M.A. Steel where we may suppose, without loss of generality, that S[e] = { { A1 U A2 }, { B1 U B2 } }. Let qf(e) := f(al)f(a2) I f(bl)f(b2). Clearly, not every tight quartet encoding is of this form; furthermore, qf does not necessarily define T. However, if f satisfies the condition f(a) E B C_C_A ==~ f(a) = f(b) for all A, B E clus(t) (3.6) then qf satisfies the condition described in Example 3.7; in particular, qf defines T. Sketch of proof It suffices to show that the edges of T can be labeled as described in Example 3.7 (so as to satisfy (3.4)) which is again achieved by induction on n := #X. Let {x, x I} be a twin in T, and suppose f({x, xl}) = x/. We define Y := X - {x} and T ~ := TIY. Then, by the condition placed on f, we obtain a corresponding function f~ : clus(t ~) --+ Y that satisfies (3.5) with f, T being replaced by y, T ~, and hence, the edges of T ~ can be ordered {el,...,en-4} so as to satisfy the condition (3.6). We then label the edge e(x, x ~) of T incident with the twin {x, x ~} as en-3, and verify that this also satisfies (3.4) for i = n This last example generalizes a result from [9] where a specific function f satisfying (3.6) is considered. The excess-free subsets of J~ for a tight quartet encoding q have a useful "patchwork" structure that we now discuss. Following [3], a collection C of subsets of a set M is called an M-patchwork if it satisfies the following condition: O#C'C_C and NC'#O ~ Nc1, uc'ec. (3.7) There are quite a number of distinct characterizations of patchworks that are ample, that is (cf. [3]), patchworks C that satisfy the condition in particular, A, BEC and #{CEC:ACCCB}=2~B-AEC, (3.8) (i) a patchwork C C_ P(M) is ample if and only if, for every cluster C E C with C I := {C E C: 0 C t C C) 0, there exist either two clusters A, B E C ~ with C = A tab, or there exists a chain C tt C C ~ with U C1~ = C; (ii) a patchwork C C_ P(M) with {m} E C for all m E M and 0, M E C is ample if and only if C contains a maximal hierarchy C, that is, a maximal subset C of P(M) for which A MB E {0, A, B) holds for all A, B E C. Lemma 3.9. Suppose q is an arbitrary quartet encoding of a binary X-tree T = (V, E; ~) and assume F1, F2 C 8. Then exc(f1 U F2) + exc(f! M F2) _< exc(f1 ) + exc(f2). (3.9)

9 Patching Up X-Trees 9 Proof In view of q(f1 M F2) C q(f1) M q(f2), we have exc(f1 to F2) + exc(f1 f3 F2) + 6 = #q(fl U F2) - #(F1 U F2) + #_q(f1 F1F2) - #(F1 n F2) < #(q(f1) U q(v2)) + #(q(rl) n q (Fz)) - (#(rl UF2) +#(Fl nfz)) = #q(fl) + #q(f2) - #F1 - #F2 = exc(f1) + exc(f2) +6. Lemma If q is a tight quartet encoding of T = (V, E; ~), then the excess-free subsets of E form a patchwork denoted by C(q). Proof Let F1, F2 C E denote subsets with exc(f1) = exc(f2) -- 0 and F1MF2 0. By Lemma 3.6(ii), we have yet by Lemma 3.9, exc(f1 M F2) >_ 0 and exc(f1 U F2) > 0, exc(f1mf2) + exc(f1uf2) < 0. Consequently, exc(ft M F2) = exc(f1 to F2) = 0, so C(q) must be a patchwork. I It is tempting to conjecture that if q defines a binary X-tree T, then there exists always an edge e E/~ such that E - {e} is q-excess-free. In some cases (e.g., for #X < 6 and for encodings constructed as in Examples 3.7 and 3.8 above), this is indeed the case, but in general it is not (Example 3.3 provides a counterexample with seven leaves). Nevertheless, we can still hope that the excess-free subsets of/~ do form at least an ample patchwork, and this turns out to be indeed the case, as we state now as part of our main result: Theorem Given a quartet encoding q of a binary X-tree T = (V, E; (p), the following three statements are equivalent: (i) q defines T; (ii) q is tight, and the patchwork C(q) of excess-free subsets of E is ample; (iii) if Q* denotes the smallest subset of Q(X) containing q(e) and containing all quartet splits ablcd for which some x E X exists with either ablcx, abldx E Q* or axlcd, ablcx E Q*, then Q* contains exactly one of the three quartet splits ablcd, actbd, or adlbc for any four distinct elements a, b, c, d E X. The implications (ii) ~ (iii) and (iii) ~ (i) follow easily from combining the lines of thought already used in the previous examples with standard results from [1] (see also [7, 8]). The implication (i) :=~ (ii), except for the fact that q must be tight, is far

10 10 S. B6cker, A.W.M. Dress, and M.A. Steel from trivial. Of course, one proceeds by induction relative to n := #X. And this allows us to assume that C(q)c_V := {F' E C(q) : F' C_ F} is an ample patchwork for all subsets F of X contained in C(q)cx := {F' E C(q) : F' C_ X and F' # X}. It then follows easily (cf. [3], Lemma 4) that max(c(q)cx) := {F E C(q)cx : F C F' E C(q)cx implies F = F'} must be a partition of X into at least three distinct subsets. The next step consists of applying the induction hypothesis to trees one derives from T by identifying pairs of twins x, y E X. This way, one is led to study decompositions of ~ into two disjoint and connected subsets F1 = F1 (x, y) and Fa --- F2(x, y) with e(x, y) E F1, #(q(fl -{e(x,y)})u{x,y}) =#Fl + 3, and exc(f2) = 0 or #(q(f2) - {x, y}) = #F Next, one shows -- and this is the most tricky part of the whole proof -- that (i) by choosing F1 = FI (x, y) maximal subject to these conditions, one can always assume exc(f2) = 0, and (ii) that exc(f2) = 0 and #F2 >_ 2 would in turn imply the existence of a connected subset F~ E C(q) with F2 C_ F~ and E - F~ E C(q) in contradiction to #max(c(q)cx) _> 3. Consequently, we can assume, for every pair x, y of twins in T, there exists a single edge f(x, y) E E - {e (x, y) } such that F1 :=/~ - {f(x, y)} and F2 := {f(x, y)} is a pair as above, that is, with #(q(,e - {e(x, y), f(x, y) } ) U {x, y} ) = #([~- {f(x, y) } ) + 3 = (#X-4)+3 = #X- 1 which in turn implies that f(x, y) must be of the form e(x ~, yl) for some pair x ~, y' of twins (because F1 =/~ - {f(x, y) } is connected) and with, say, x r the unique element in X missing in q_(/~ - {e(x, y), f(x, y)}). Applying the same argument now to the twins x', yt and repeating this process iteratively will therefore eventually produce a sequence of distinct pairs xb yl; x2, Y2;... ; xk, Yk of twins such that, for i = 1,...,k (mod k), we have q(l~ -- {e(xi, Yi), e(xi+l, Yi+I)}) U {xi, Yi} = X - {Xi+l } which in turn implies easily that we can construct a second X-tree (W, Et; ~) that is concordant with q(e): Note first that our assumptions imply that, for each i = 1,...,k, there must exist some zi E X with xiyilxi+lzi = q(e(xi, yi)), and that zi E {Xl,...,xk} cannot hold because, according to our construction, e(xi, yi) is the only edge in E - {e(xi+l,yi+l)} with Xi+l E q.(e(xi,yi)). Hence, we can cut off, for every i = 1,...,k

11 Patching Up X-Trees t 1 Xi+l //-- Yi+l ~--- Yi+l Yi % Yi % "% Xi+lZi Figure 3: Cutting off and re-implanting edges. (mod k), the edge incident with d~(xi+l) and implant it instead into the edge incident with ~(zi) as depicted in Figure 3. Ifzi = zi+l, we have to make sure that the edge leading to ~(xi+2) is implanted closer to O(zi) than the edge leading to t~(xi+l); no further special care needs to be taken (and because we cannot have zt = z2... Zk, this requirement can always be fulfilled). Finally, defining W, E ~ and t~ I accordingly, we find a second X-tree (W, E~; t~ ~) in T(q(E)) that is clearly non-isomorphic with (V, E; ~)), the final contradiction. Remark. Note that, for the tree T1 considered in Example 3.2 (see also Figure 1), we can choose f(2i- 1, 2i) E {e j, ek} for all {i, j, k} = {1, 2, 3}; so we can use the twin sequence 1, 2; 4, 3 as well as the twin sequence 2, 1; 4, 3; 6, 5, leading to Zl = 5 and z or Zl = 5, z2 = 1, and z3 = 3, respectively; indeed, both rearrangements lead to a tree isomorphic to T2. The detailed proof will take up more than 25 pages of reasoning, and so it will be published elsewhere (see [4]). Corollary Suppose a set of quartet splits Q with #Q = #Q_Q- 3 > 2 is concordant with a unique tree. Then Q is the disjoint union of two proper subsets Q1 and Q2 with #Qi = #Q 2i - 3 for i = 1, 2 such that #T( Qi) = 1 holds for i = 1,2. It is worth mentioning that the above theorem implies the result mentioned in Example 3.4. To this end, let q : ~ ~ Q(X) denote a tight quartet encoding of a binary X-tree T = (V, E; t~). Suppose condition (3.2) is satisfied. We assume x E NQcQQ, define v < u for u, v E V if the path from ~(x) to u passes through v, and put V(v):={uEV: v<_u} and /~(v):={eee: ec_v(v)}. Now, it is easy to see that C I := {E(v) : v E V} is a maximal/~-hierarchy, and that every non-empty element from C r is in C(q). In view of the results in [3], this implies that C(q) is ample which in turn, according to Theorem 3.11, implies that q defines T. Applications regarding tree amalgamation algorithms will be discussed in a separate paper [2].

12 12 S. BOcker, A.W.M. Dress, and M.A. Steel References 1. H.-L Bandelt and A. Dress, Reconstructing the shape of a tree from observed dissimilarity data, Adv. in Appl. Math. 7(3) (1986) S. BOcker, D. Bryant, A.W. Dress, and M. Steel, Algorithmic aspects of tree amalgamation, in preparation. 3. S. B6cker and A.W. Dress, Patchworks, 1999, to appear in Adv. in Math. 4. S. Brcker, A.W. Dress, and M. Steel, Most parsimonious quartet encodings of binary trees, in preparation. 5. P. Buneman, The recovery of trees from measures of dissimilarity, In: Mathematics in the Archaeological and Historical Sciences, F. Hodson, D. Kendall, and P. Tautu, Eds., Edinburgh University Press, Edinburgh, 1971, pp R.L Cann, M. Stoneking, and A.C. Wilson, Mitochondrial DNA and human evolution, Nature 325 (1987) , H. Colonius and H.-H. Schulze, Tree structures for proximity data, British J. Math. Statist. Psych. 34(2) (1981) H. Colonius and H.-H. Schulze, Repr~isentation nichtnumerischer.~hnlichkeitsdaten durch Baumstrukturen, Psych. Beitr. 21 (1979) P.L. Erd6s, L.A. Szrkely, M. Steel, and T. Wamow, A few logs suffice to build (almost) all trees, Random Structures Algorithms 14 (1999) D. Huson, S. Nettles, L. Parida, T. Wamow, and S. Yooseph, The disk-covering method for tree reconstruction, In: Proceedings of "Algorithms and Experiments" (ALEX98), Trento, Italy, 9-11 February, 1998, R. Battiti and A. Bertossi, Eds., 1998, pp Available from WWW site science, unitn, it/alex98/proceedings, html. 11. D. Huson and T.J. Warnow, Obtaining highly accurate topology and evolutionary estimates of evolutionary trees from very short sequences, RECOMB 99, to appear. 12. P.M. Robinson, Computer-assisted stemmatic analysis and 'best-text' historical editing, In: Studies in Stemmatology, P. Reenen and M. van Mulken, Eds., John Benjamins Publishing, Amsterdam, 1996, pp E. SchrOder, Vier Kombinatorische Probleme, Z. Math. Phys. 15 (1870) t4. M. Steel, The complexity of reconstructing trees from qualitative characters and subtrees, J. Classification 9(1) (1992) M. Steel, A.W.M, Dress, and S. BOcker, Some simple but fundamental limitations on supertree and consensus tree methods, 1999, submitted. 16. K. Strimmer and A. von Haeseler, Quartet puzzling: A quartet maximum likelihood method for reconstructing tree topologies, Mol. Biol. Evol. 13(7) (1996) D.L. Swofford, G.J. Olsen, P.J. Waddell, and D.M. Hillis, Phylogenetic inference, In: Molecular Systematics, 2nd Ed., D. Hillis, C. Moritz, and B. Marble, Eds., Sinauer Associates, second ed., 1996, pp T. Warnow, Mathematical approaches to comparative linguistics, Proc. Nat. Acad. Sci. U.S.A. 94(13) (1997) S.J. Witlson, Measuring inconsistency in phylogenetic trees, J. Theor. Biol. 190(1) (1998)