Recovering Sdr Trees from Symbolic

A well known result from cluster theory states that there is a 1-to-1 correspondence between dated, compact, rooted trees and ultrametrics. In this paper, we generalize this result yielding a canonical 1-to-1 correspondence between symbolically dated trees and symbolic ultrametrics, using an arbitrary set as the set of (possible) dates or values. It turns out that a rather unexpected new condition is needed to properly deene symbolic ultrametrics so that the above correspondence holds. In the second part of the paper, we use our main result to derive, as a corollary, a theorem by H. J. Bandelt and M. A. Steel regarding a canonical 1-to-1 correspondence between additive trees and metrics satisfying the 4-point condition, both taking their values in abelian monoids. All (di-)graphs G = (V; E V 2) studied in this paper will be nite (and { by deenition { without multiple edges). For a vertex v, let d ? (v) := #fw 2 V : (w; v) 2 Eg denote its in-degree, and d + (v) := #fw 2 V : (v; w) 2 Eg its out-degree. A path u 0 u 1 : : : u l of length l 0 is a sequence of vertices u 0 ; u 1 ; : : : ; u l 2 V , such that (u j?1 ; u j) 2 E for j = 1; : : : ; l. A rooted tree T = (V; E) is a connected digraph (that is, the associated undirected graph is connected) such that there exists exactly one vertex r 2 V (the root) with d ? (r) = 0 while we have d ? (v) = 1 for all v 2 V ?frg. The leaves of a rooted tree are the vertices v of out-degree 0, all other vertices are called inner vertices. A rooted tree is called compact if d + (v) 2 holds for all inner vertices v. In a rooted tree T = (V; E), the last common ancestor lca(u; v) = lca T (u; v) of u; v 2 V is holds if k l and u = v k , lca(u; v) = v if l k and v = u l , and lca(u; v) = u j if u j = v j and u j+1 6 = v j+1 holds for some j < minfk; lg. Note also that a rooted …