We use the Mass Transport Principle to analyze the local recursion governing the resolvent (A−z) −1 of the adjacency operator of unimodular random trees. In the limit where the complex parameter z approaches a given location λ on the real axis, we show that this recursion induces a decomposition of the tree into finite blocks whose geometry directly determines the spectral mass at λ. We then exploit this correspondence to obtain precise information on the pure-point support of the spectrum, in terms of expansion properties of the tree. In particular, we deduce that the pure-point support of the spectrum of any unimodular random tree with minimum degree δ ≥ 3 and maximum degree ∆ is restricted to finitely many points, namely the eigenvalues of trees of size less than ∆−2 δ−2. More generally, we show that the restriction δ ≥ 3 can be weakened to δ ≥ 2, as long as the anchored isoperimetric constant of the tree remains bounded away from 0. This applies in particular to any unimodular Galton-Watson tree without leaves, allowing us to settle a conjecture of Bordenave, Sen and Virág (2013).