The endpoints of a longest path in a finite tree must have degree one. The point is to make the formal expression readable enough to audit line by line.
Definitions first
A finite graph \(T=(V,E)\) is a tree when it is connected and acyclic. The unique simple path \(P_{uv}\) between two vertices makes distance and recursive decomposition especially rigid.
The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.
A small case in full
The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.
The reusable statement
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
A nearby false statement
A rooted tree adds a parent relation that an unrooted tree does not possess. Statements about ancestors depend on the chosen root.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.