Electrical treeing is one of the main degradation mechanisms in high-voltage polymeric insulation. Epoxy resin is used as insulating material in power equipment such as rotating machines, power transformers, gas-insulated switchgears, and insulators, among others. Electrical trees grow under the effect of partial discharges (PDs) that progressively degrade the polymer until the tree crosses the bulk insulation, then causing the failure of power equipment and the outage of the energy supply. This work studies electrical trees in epoxy resin through different PD analysis techniques, evaluating and comparing their ability to identify tree bulk-insulation crossing, the precursor of failure. Two PD measurement systems were used simultaneously—one to capture the sequence of PD pulses and another to acquire PD pulse waveforms—and four PD analysis techniques were deployed. Phase-resolved PD (PRPD) and pulse sequence analysis (PSA) identified tree crossing; however, they were more sensible to the AC excitation voltage amplitude and frequency. Nonlinear time series analysis (NLTSA) characteristics were evaluated through the correlation dimension, showing a reduction from pre- to post-crossing, and thus representing a change to a less complex dynamical system. The PD pulse waveform parameters had the best performance; they could identify tree crossing in epoxy resin material independently of the applied AC voltage amplitude and frequency, making them more robust for a broader range of situations, and thus, they can be exploited as a diagnostic tool for the asset management of high-voltage polymeric insulation.