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A number of hook formulas and hook summation formulas have previouslyappeared, involving various classes of trees. One of these classes of trees isrooted trees with labelled vertices, in which the labels increase along everychain from the root vertex to a leaf. In this paper we give a new hooksummation formula for these (unordered increasing) trees, by introducing a newset of indeterminates indexed by pairs of vertices, that we call edge weights.This new result generalizes a previous result by F\'eray and Goulden, thatarose in the context of representations of the symmetric group via the study ofKerov's character polynomials. Our proof is by means of a combinatorialbijection that is a generalization of the Pr\"ufer code for labelled trees.

An edge-weighted hook formula for labelled trees