摘要翻译：
设T是一棵有n个叶子的加权树。设D_{i,j}为叶子i和j之间的距离。设D_{i,j,k}=(D_{i,j}+D_{j,k}+D_{i,k}）/2。我们将这样的数字称为树的“三重”。本文给出了一个不同于以往的刻画，即由$n$-集的2-子集索引的集是树的双权。利用同样的思想，我们还得到了由$n$-集的3-子集索引的实数集是具有$n$-叶的树的三重权集的充要条件。此外，我们还对Saitou-Nei的邻域连接算法进行了改进，以从数据D_{i,j}中重建树。
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英文标题：
《Sets of double and triple weights of trees》
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作者：
Elena Rubei
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最新提交年份：
2011
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Combinatorics        组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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英文摘要：
  Let T be a weighted tree with n leaves. Let D_{i,j} be the distance between the leaves i and j. Let D_{i,j,k}= (D_{i,j} + D_{j,k} +D_{i,k})/2. We will call such numbers "triple weights" of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a $n$-set to be double weights of a tree. By using the same ideas,we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an $n$-set to be the set of the triple weights of a tree with $n$ leaves. Besides we propose a slight modification of Saitou-Nei's Neighbour-Joining algorithm to reconstruct trees from the data D_{i,j}.
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PDF链接：
https://arxiv.org/pdf/0712.3026