Subgraphs smaller than the girth

For graphs A, B, let ( BA) denote the number of subsets of nodes of A for which the induced subgraph is B. If G and H both have girth > k, and if ( TG) = ( TH) for every k-node tree T, then for every k-node forest F, ( FG) = ( GH). Say the spread of a tree is the number of nodes in a longest path. If G is regular of degree d, on n nodes, with girth > k, and if F is a forest of total spread <=k, then the value of ( FG) depends only on n and d.