Bounds on isoperimetric values of trees

Let G=(V,E) be a finite, simple and undirected graph. For S@?V, let @d(S,G)={(u,v)@?E:u@?S and v@?V-S} be the edge boundary of S. Given an integer i, 1@?i@?|V|, let the edge isoperimetric value of G at i be defined as b"e(i,G)=min"S"@?"V";"|"S"|"="i|@d(S,G)|. The edge isoperimetric peak of G is defined as b"e(G)=max"1"@?"j"@?"|"V"|b"e(j,G). Let b"v(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees. The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as T"d^2), c"1d@?b"e(T"d^2)@?d and c"2d@?b"v(T"d^2)@?d where c"1, c"2 are constants. For a complete t-ary tree of depth d (denoted as T"d^t) and d>=clogt where c is a constant, we show that c"1td@?b"e(T"d^t)@?td and c"2dt@?b"v(T"d^t)@?d where c"1, c"2 are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T=(V,E,r) be a finite, connected and rooted tree - the root being the vertex r. Define a weight function w:V->N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index @h(T) be defined as the number of distinct weights in the tree, i.e @h(T)=|{w(u):u@?V}|. For a positive integer k, let @?(k)=|{i@?N:1@?i@?|V|,b"e(i,G)@?k}|. We show that @?(k)@?22@h+kk.