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The increasing use of phylogeny in biological studies is limited by the need to make available more efficient tools for computing distances between trees. The geodesic tree distance-introduced by Billera, Holmes, and Vogtmann-combines both the tree topology and edge lengths into a single metric. Despite the conceptual simplicity of the geodesic tree distance, algorithms to compute it don't scale well to large, real-world phylogenetic trees composed of hundred or even thousand leaves. In this paper, we propose the geodesic distance as an effective tool for exploring the likelihood profile in the space of phylogenetic trees, and we give a cubic time algorithm, GeoHeuristic, in order to compute an approximation of the distance. We compare it with the GTP algorithm, which calculates the exact distance, and the cone path length, which is another approximation, showing that GeoHeuristic achieves a quite good trade-off between accuracy (relative error always lower than 0.0001) and efficiency. We also prove the equivalence among GeoHeuristic, cone path, and Robinson-Foulds distances when assuming branch lengths equal to unity and we show empirically that, under this restriction, these distances are almost always equal to the actual geodesic.