  For an arbitrary valued field $(K,v)$ and a given extension
$v(K^*)\hookrightarrow\Lambda$ of ordered groups, we analyze the structure of
the tree formed by all $\Lambda$-valued extensions of $v$ to the polynomial
ring $K[x]$. As an application, we find a model for the tree of all equivalence
classes of valuations on $K[x]$ (without fixing their value group), whose
restriction to $K$ is equivalent to $v$.
  In the henselian case, we apply these results to show that there is a
complete parallelism between the arithmetic properties of irreducible
polynomials $F\in K[x]$, encoded by their Okutsu frames, and the
valuation-theoretic properties of their induced valuations $v_F$ on $K[x]$,
encoded by their MacLane-Vaqui\'e chains. This parallelism was only known for
defectless irreducible polynomials.
