Generalized tree alignment

In computational phylogenetics, generalized tree alignment is the problem of producing a multiple sequence alignment and a phylogenetic tree on a set of sequences simultaneously, as opposed to separately.^[1]

Formally, Generalized tree alignment is the following optimization problem.

Input: A set ${\displaystyle S}$ and an edit distance function ${\displaystyle d}$ between sequences,

Output: A tree ${\displaystyle T}$ leaf-labeled by ${\displaystyle S}$ and labeled with sequences at the internal nodes, such that ${\displaystyle \Sigma _{e\in T}d(e)}$ is minimized, where ${\displaystyle d(e)}$ is the edit distance between the endpoints of ${\displaystyle e}$.^[2]

Note that this is in contrast to tree alignment, where the tree is provided as input.