The second largest eigenvalue of a tree

Denote by λ₂(T) the second largest eigenvalue of a tree T. An easy algorithm is given to decide whether λ₂(T)?λ for a given number λ, and a structure theorem for trees withλ₂(T)?λ is proved. Also, it is shown that a tree T with n vertices has $\text{λ}{}_2\text{(T)?lsqb}\text{(n?3)}\text{2}\text{rsqb}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$; this bound is best possible for odd n.