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The complete set of Eigenstates and Eigenvalues of the nearest neighbourtight binding model on a Cayley tree with branching number $b=2$ and $M$branching generations with open boundary conditions is derived. We find that ofthe $N= 1 +3 (2^M-1)$ total states only $3 M +1$ states are extended throughoutthe Cayley tree. The remaining $N-(3 M+1)$ states are found to be stronglylocalised states with finite amplitudes on only a subset of sites. Inparticular, there are, for $M>1$, $3 \times 2^{M-2}$ surface states which areeach antisymmetric combinations of only two sites on the surface of the Cayleytree and have energy eactly at $E=0$, the middle of the band. The ground stateand the first two excited states of the Cayley tree are found to be extendedstates with amplitudes on all sites of the Cayley tree, for all $M$. We use theresults on the complete set of Eigenstates and Eigenvalues to derive the totaldensity of states and a local density of states.

Complete Solution of the Tight Binding Model on a Cayley Tree: Strongly Localised versus Extended States