The success of future organic electronic devices distinctively depends on the electronic and geometric properties of thin organic films. Although obviously these properties are strongly influenced by the growth mechanisms, real time growth studies are relatively rare since not many experimental techniques exist that allow in situ studies in ultra high vacuum. In this context, we investigated the prototypical system 1,4,5,8-naphtalene-tetracarboxylic-dianhydride (NTCDA) on Cu(001). We used low-energy electron microscopy (LEEM) for the real-time growth study, and a variety of other techniques for investigating the geometric and electronic structure. While for similar model systems well known and well characterized growth modi occur (e.g., compact, well ordered islands or disordered, gas-like layers), for NTCDA/Cu(001) we observe the growth of dendrite-like, fractal structures. The dendritic structures arise from a strongly preferred one-dimensional growth mode forming a long-range ordered network of thin molecular chains spanning over the entire surface already at small coverages. Later in the growth process, the voids in the network structure are incrementally filled. These results are very unexpected for such a simple adsorbate system consisting of well investigated components, the properties of which were believed to be already well understood. We explain this unexpected behavior by a dendritic growth model that is supported by energetic arguments based on pair-potential calculations. These calculations give reason for the experimentally observed growth of one-dimensional structures, and therefore represent the key to a semi-quantitative understanding of this dendritic growth mode.