  The uniform recursive tree (URT) is one of the most important models and has
been successfully applied to many fields. Here we study exactly the topological
characteristics and spectral properties of the Laplacian matrix of a
deterministic uniform recursive tree, which is a deterministic version of URT.
Firstly, from the perspective of complex networks, we determine the main
structural characteristics of the deterministic tree. The obtained vigorous
results show that the network has an exponential degree distribution, small
average path length, power-law distribution of node betweenness, and positive
degree-degree correlations. Then we determine the complete Laplacian spectra
(eigenvalues) and their corresponding eigenvectors of the considered graph.
Interestingly, all the Laplacian eigenvalues are distinct.
