The graph edit distance is used for comparing graphs in various domains. Due to its high computational complexity it is primarily approximated. Widely-used heuristics search for an optimal assignment of vertices based on the distance between local substructures. While faster ones only consider vertices and their incident edges, leading to poor accuracy, other approaches require computationally intense exact distance computations between subgraphs. Our new method abstracts local substructures to neighborhood trees and compares them using efficient tree matching techniques. This results in a ground distance for mapping vertices that yields high quality approximations of the graph edit distance. By limiting the maximum tree height, our method supports steering between more accurate results and faster execution. We thoroughly analyze the running time of the tree matching method and propose several techniques to accelerate computation in practice. We use compressed tree representations, recognize redundancies by tree canonization and exploit them via caching. Experimentally we show that our method provides a significantly improved trade-off between running time and approximation quality compared to existing state-of-the-art approaches.