  To each link $L$ in $S^3$ we associate a collection of certain labelled
directed trees, called width trees. We interpret some classical and new
topological link invariants in terms of these width trees and show how the
geometric structure of the width trees can bound the values of these invariants
from below. We also show that each width tree is associated with a knot in
$S^3$ and that if it also meets a high enough "distance threshold" it is, up to
a certain equivalence, the unique width tree realizing the invariants.
