Let be a decreasing family of connected, open sets in R2 such that the limit lim is a tree γ. We shall discuss in what sense the Neumann Laplacian Hω on fie tends to an operator on γ. After giving a general formula which applies to any tree T (Theorem 4.5), we shall show that a more concrete formula holds for some specific domains (Example 4.8 and §5), where γ is the trace operator from H1 (ω), ω = ω1, into a weighted L2 space on the tree γ, z ∈ C\[0, ), and, for a function f on ω, fe is the restriction of g on ω. © 2001, Oldenbourg Wissenschaftsverlag GmbH, Rosenheimer Str. 145, 81671 München. All rights reserved.