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To each link $L$ in $S^3$ we associate a collection of certain labelleddirected trees, called width trees. We interpret some classical and newtopological link invariants in terms of these width trees and show how thegeometric structure of the width trees can bound the values of these invariantsfrom 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 toa certain equivalence, the unique width tree realizing the invariants.

Links, bridge number, and width trees