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 k , T 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.