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 [9,6], 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 the present work is based on the truncation of the meromorphic series that approximate the symbol of the Dirichlet-to-Neumann operator, in the spirit of the absorbing boundary conditions of B. Engquist and A. Majda. We analyze its stability and convergence, as well as present computational aspects of the method. Numerical results confirm theoretical findings.