  We define an algorithm k which takes a connected graph G on a totally ordered
vertex set and returns an increasing tree R (which is not necessarily a subtree
of G). We characterize the set of graphs G such that k(G)=R. Because this set
has a simple structure (it is isomorphic to a product of non-empty power sets),
it is easy to evaluate certain graph invariants in terms of increasing trees.
In particular, we prove that, up to sign, the coefficient of x^q in the
chromatic polynomial of G is the number of increasing forests with q components
that satisfy a condition that we call G-connectedness. We also find a bijection
between increasing G-connected trees and broken circuit free subtrees of G.
