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Linear structural causal models (SCMs) are used to analyze the relationships between random variables. Directed edges represent direct causal effects and bidirected edges represent hidden confounders. Generically identifying the causal parameters from observed correlations between the random variables is an open problem in causality. Gupta and Bläser solve the case of SCMs in which the directed edges form a tree by giving a randomized polynomial time algorithm with running time $O(n^6)$. We present an improved algorithm with running time $O(n^3 \log^2 n)$ and demonstrate its feasibility by providing an implementation that outperforms existing state-of-the-art implementations.