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0, 0). Hence, employing ∞ ϕ′n;j (0)ϕ′n;k (0) n=1 (λn − λ)m+1 provided m ≥ 2. , the Dirichlet eigenvalues and the Neumann data of the Dirichlet eigenfunc˜ (m) . tions. Therefore the exact same expression is obtained for Λ j,k ˜ tends to zero. 2. As λ tends to negative infinity Λ − Λ Proof. 1 the quantities Λj,k = −ψj′ (k, λ, 0) are exponentially small √ except for j = k in√which case we have Λk,k = ψk′ (k, λ, 0) = − −λ + o(1). But ˜ k,k = o(1). ˜ k,k = − −λ + o(1), we have that (Λ − Λ) since also Λ On Inverse Problems for Finite Trees 45 7.