Hierarchical clustering (HC) algorithms are generally limited to small data instances due to their runtime costs. Here we mitigate this shortcoming and explore fast HC algorithms based on random projections for single (SLC) and average (ALC) linkage clustering as well as for the minimum spanning tree problem (MST). We present a thorough adaptive analysis of our algorithms that improve prior work from $O(N^2)$ by up to a factor of $N/(\log N)^2$ for a dataset of $N$ points in Euclidean space. The algorithms maintain, with arbitrary high probability, the outcome of hierarchical clustering as well as the worst-case running-time guarantees. We also present parameter-free instances of our algorithms.