This work is dedicated to the construction and analysis of high-order transparent boundary conditions for the weighted wave equation on a fractal tree, which models sound propagation inside human lungs. This article follows the works [10, 9], aimed at the analysis and numerical treatment of the model, as well as the construction of low-order and exact discrete boundary conditions. The method suggested in this article is based on the truncation of the meromorphic series that approximate the symbol of the Dirichlet-to-Neumann operator, similarly to the absorbing boundary conditions of B. En-gquist and A. Majda. We analyze its stability, convergence and complexity. The error analysis is largely based on spectral estimates of the underlying weighted Laplacian. Numerical results confirm the efficiency of the method.