We study the spin 1/2 quantum Heisenberg antiferromagnet on a Bethe lattice diluted to the percolation threshold. Dilution creates areas of even/odd sublattice imbalance resulting in "dangling spins'' : spins which can not be paired up into dimers with their nearest neighbors (L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006); Phys. Rev. B 81, 054417 (2010)). These spins are found to significantly affect the low energy spectrum: the lowest energy gaps are found to scale 'anomalously' with system size (N), being much smaller than the expected 1/N scaling. By computing multiple excited states in the low energy spectrum (using a version of the Density Matrix Renormalization Group (DMRG) algorithm for generic tree graphs), and studying their correspondence with the geometry of the percolation cluster, we provide evidence that the dangling spins collectively act as "emergent'' spin 1/2 (or in some cases spin-1) degrees of freedom. By evaluating inter-site correlation and susceptibility matrices, we detect the presence and location of these emergent spins. We process our numerical data to compute an effective Hamiltonian in terms of the emergent spins, and find them to have pairwise Heisenberg interactions that decay exponentially with distance.