If $B$ is a branch at $O\in\mathbb{C}^2$ of a holomorphic curve, a
Puiseux parametrisation $y=\psi(x)$ of $B$ determines "pro-branches"
defined over a sector $|\mathrm{arg} x-\alpha| < \varepsilon $.
The exponent of contact of two pro-branches is the (fractional)
exponent of the first power of $x$ where they differ. We first
show how to use exponents of contact to give simple proofs of
several well known results.
For $C$ the germ at $O$ of a curve in $\mathbb{C}^2$, the Eggers tree
$T_C$ of $C$ is defined. We also introduce combinatorial
invariants (particularly, a certain 1-chain) on $T_C$. Any other
germ $\Gamma$ at $O$ has contact with $C$ measured by a unique point
$X_{\Gamma}\in T_C$, and this determines the set of exponents of
contact with $C$ of any pro-branch of $\Gamma$. A simple formula
establishes the converse, and this leads to a short proof of the
theorem on decomposition of a transverse polar of $C$ into
parts $P_i$, where both the multiplicity of $P_i$, and the order
of contact with $C$ of each branch $Q$ of $P_i$ are explicitly
given.