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Székely László: New distances between phylogenetic trees

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Monday, 19. May 2025, 16:00 - 16:30
A phylogenetic tree with n leaves is a  tree, in which  every degree is 1 or 3, the degree 1 vertices are labeled bijectively with 1,2,...,n, and the degree 3 vertices are unlabeled.
Distance concepts between phylogenetic trees with n leaves are very relevant,  if one wants to speak  about 'almost correct' phylogeny reconstruction.
Two standard distance concepts are the Robinson-Foulds distance and the quartet distance. 
There has been great interest in the fast computation of tthe quartet distance.  
A 40 years old Bandelt-Dress conjecture is about the asymptotics of the diameter of the tree space in quartet distance.

We introduce a family of $d_i$ distances between phylogenetic trees with n leaves. 
These are natural generalizations of the previously mentioned distances, such that $d_2$ is the  Robinson-Foulds distance and $d_{n-2}$ is the quartet distance.
A $k$-partition of  $\{1,2,...,n\}$    is induced by a phylogenetic tree with $n$ leaves, if after the removal of some $k-1$ edges of the tree, the partition classes are each living on a different tree of the
remaining forest. 
The $d_k$ distance between two phylogenetic trees with n leaves depends linearly on the number of common induced $k$-partitions of them.
(The  induced partitions are often called convex characters, and the common induced partitions   biconvex characters in the phylogenetic literature.)

We proved a number of extremal results for the least number of common  k-partitions (or partitions) of two phylogenetic  trees with n leaves (or two phylogenetic caterpillar trees with $n$ leaves). 
Bounds on these quantities translate into bounds for the diameter of the tree space (or caterpillar tree space)  for the $d_i$ distance.

We extended some results for the least number of common induced k-partitions (or partitions) from 2  to m phylogenetic  trees with n leaves.

This work was done when EC, VM and LAS were in residence at ICERM, Brown University, at the semester program ''Theory, Methods, and Applications of Quantitative Phylogenomics'', Fall 2024. Joint work with Éva Czabarka, University of South Carolina, Steven Kelk,  Maastricht University, Vincent Mouton, University of East England.

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