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.