B. Zelinka [Zel] showed that the median of any tree is equal to its centroid. A. Kang and D. Ault [Kang] extended this result for any tree with weights on its edges. In Chapter 2, we extend the result further to any tree with
weights on its edges and vertices and show that the second median of any tree with weights on its vertices is equal to its second centroid. The main result in Chapter 3 is that if S is a spider, T is a tree and S , T have the same status sequence,than S is isomorphic to T . In Chapter 4, we show that if T is a weakly
status-injective tree, T’’ is a tree, and T ,T’’ have thesame status sequence, then T is isomorphic to T’’. The main result in Chapter 5 is that if two spiders have the same branch-weight-sequence, then they are isomorphic.
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