  Consider a germ $C$ of reduced curve on a smooth germ $S$ of complex analytic
surface. Assume that $C$ contains a smooth branch $L$. Using the Newton-Puiseux
series of $C$ relative to any coordinate system $(x,y)$ on $S$ such that $L$ is
the $y$-axis, one may define the {\em Eggers-Wall tree} $\Theta_L(C)$ of $C$
relative to $L$. Its ends are labeled by the branches of $C$ and it is endowed
with three natural functions measuring the characteristic exponents of the
previous Newton-Puiseux series, their denominators and contact orders. The main
objective of this paper is to embed canonically $\Theta_L(C)$ into Favre and
Jonsson's valuative tree $\mathbb{P}(\mathcal{V})$ of real-valued
semivaluations of $S$ up to scalar multiplication, and to show that this
embedding identifies the three natural functions on $\Theta_L(C)$ as pullbacks
of other naturally defined functions on $\mathbb{P}(\mathcal{V})$. As a
consequence, we prove an inversion theorem generalizing the well-known
Abhyankar-Zariski inversion theorem concerning one branch: if $L'$ is a second
smooth branch of $C$, then the valuative embeddings of the Eggers-Wall trees
$\Theta_{L'}(C)$ and $\Theta_L(C)$ identify them canonically, their associated
triples of functions being easily expressible in terms of each other. We prove
also that the space $\mathbb{P}(\mathcal{V})$ is the projective limit of
Eggers-Wall trees over all choices of curves $C$. As a supplementary result, we
explain how to pass from $\Theta_L(C)$ to an associated splice diagram.
