Comparative analyses of phylogenetic trees typically require identical taxon sets, however, in practice, trees often include distinct but overlapping taxa. Pruning non-shared leaves discards phylogenetic signal, whereas tree completion can preserve both taxa and branch-length information. This work introduces a polynomial-time algorithm for set-wide completion of phylogenetic trees with partial taxon overlap. The proposed method identifies and extracts maximal completion subtrees that frequently appear across the source trees and constructs a weighted majority-rule consensus. Branch lengths are scaled using rates derived from common leaves. Each consensus subtree is inserted at the position that minimizes the quadratic distance error measured against information from the source trees, with candidate positions restricted to the original branches of the target tree. We demonstrate that the algorithm runs in polynomial time and preserves distances among the original taxa, yielding a unique completion that is order-independent with respect to the processing order of target trees. An experimental evaluation on amphibians, mammals, sharks, and squamates shows that the proposed method consistently achieves the lowest distance to the subset reference trees across subsets among all methods, in both topology and branch lengths. An open-source Python implementation of the proposed algorithm and the biological datasets utilized in this study are publicly available at: https://github.com/tahiri-lab/overlap-treeset-completion/.

翻译：系统发育树的比较分析通常需要相同分类单元集合，然而在实践中，树往往包含不同但重叠的分类单元。剪除非共有叶片会丢失系统发育信号，而树补全能够同时保留分类单元和分支长度信息。本研究提出一种多项式时间算法，用于对存在部分分类单元重叠的系统发育树进行集合级补全。该方法识别并提取源树中频繁出现的最大补全子树，并构建加权多数规则一致性树。分支长度通过从共有叶片推断的速率进行缩放。每个一致性子树被插入到最小化与源树信息二次距离误差的位置，候选位置限定于目标树的原始分支。我们证明该算法在多项式时间内运行，能保持原始分类单元间的距离，并生成与目标树处理顺序无关的唯一补全结果。在两栖类、哺乳类、鲨鱼和鳞龙类上的实验评估表明，无论在拓扑结构还是分支长度上，该方法在所有方法中始终获得与子集参考树的最低距离。本算法及所用生物数据集的开源Python实现已公开于：https://github.com/tahiri-lab/overlap-treeset-completion/。