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As before, Ω x) is an open formula with x as its only free variable; Ω(m) is the result of replacing the free instances of x with a name for m.
Herein, we use that every path rooted at one of its endpoints results in a cross-pair-free instance of MTO.
Informally speaking, a cross-pair-free instance only contains source target pairs whose corresponding paths are directed either towards the root or towards the leaves, but do not change their direction.
The Generalized Horn Flat patterns have the following form: (a1 > b1), …, (an > bn) ⊢ (a > b); where negations are allowed in the left hand side of the pattern and where the letters ai, a, etc only admit conditional-free instances.
Cross-pair-free instances of W-MTO are of special interest since they constitute our "trivial instances".
Now, we devise a dynamic-programming-based algorithm solving W-MTO in quadratic time on cross-pair-free instances.
To this end, we introduce the concept of "cross pairs" and show that for cross-pair-free instances MTO can be solved in O(n) time.
Next, we show that W-MTO is fixed-parameter tractable with respect to the parameter q v by extending the dynamic programming algorithm for cross-pair-free instances.
Theorem 2. On n-vertex trees, W EIGHTED M AXIMUM T REE O RIENTATION for cross-pair-free instances with given root can be solved in O(n) time.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com