On multiple eigenvalues of trees

Let T be a tree of order n>6 with μ as a positive eigenvalue of multiplicity k. Star complements are used to show that (i) if k>n/3 then μ=1, (ii) if μ=1 then, without restriction on has k+1 pendant edges that form an induced matching. The results are used to identify the trees with a non-zero eigenvalue of maximum possible multiplicity.