  As one of the most significant models, the uniform recursive tree (URT) has
found many applications in a variety of fields. In this paper, we study
rigorously the structural features and spectral properties of the adjacency
matrix for a family of deterministic uniform recursive trees (DURTs) that are
deterministic versions of URT. Firstly, from the perspective of complex
networks, we investigate analytically the main structural characteristics of
DURTs, and obtain the accurate solutions for these properties, which include
degree distribution, average path length, distribution of node betweenness, and
degree correlations. Then we determine the complete eigenvalues and their
corresponding eigenvectors of the adjacency matrix for DURTs. Our research may
shed light in better understanding of the features for URT. Also, the
analytical methods used here is capable of extending to many other
deterministic networks, making the precise computation of their properties
(especially the full spectrum characteristics) possible.
