On the number of trees having k edges in common with a graph of bounded degrees

In this paper it is proved that for any fixed r and k the number T(G; n, k) of spanning trees of Kn having k edges in common with a fixed subgraph G of Kn having s edges and degrees not larger than r, satisfies limn → ∞T(G; n, k)/nn − 2 = (2λ)k e−2λ/k! where λ = limn → ∞ s/n. This solves a conjecture raised in Tomescu (1985).