Abstract
The authors study the total numbers of lattice trees with specified topologies. For strongly embeddable (or site) clusters with ni branching points of degree i, they show how to prove rigorously that the growth constants exist and are all equal to the neighbour-avoiding walk limit nu . They derive some exact upper bounds for the critical exponents associated with the 'star', 'comb' and 'brush' topologies. Exact enumeration data are derived and analysed for both weak and strong embeddings of some stars, combs and brushes on the square, triangular, simple cubic and d=4 simple hypercubic lattices. Using the exact enumeration data for the general d-dimensional simple hypercubic lattice and the exact results for the interior of a Bethe lattice, they derive expansions for the growth constants in inverse powers of the dimensionality. These results are consistent with the growth constants being equal to the appropriate walk limits ( mu or nu ).
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