A 2-tree is the case 
of a k-tree. It is obtained from a triangle
graph by repeatedly adding a new graph vertex adjacent to exactly the two ends of an existing
graph edge. A 2-tree on 
vertices has 
edges and is a planar
graph. An alternate convention starts from a single graph
edge and also includes 
(Sabaei et al. 2026).
Let 
be a 2-tree on 
vertices, and let 
be the infimum of the ratio
of longest to shortest graph edge length over all planar
straight line drawings of 
. Sabaei et al. (2026) proved
 |
where 
is the floor
function. Together with the previously known matching lower bound, this shows
that the largest necessary ratio among 
-vertex 2-trees is 
, where 
denotes big-Theta notation.
The bound concerns an infimum and does not require a
drawing attaining it.
