Quadtree is a hierarchical data structure that is well-suited for h-adaptive mesh refinement. Due to the presence of hanging nodes, classical shape functions are non-conforming on quadtree meshes. In this paper, we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes. To this end, the recently proposed construction of polygonal basis functions is adapted to quadtree elements. A fast technique for calculating stiffness matrix on quadtree meshes is introduced. Residual-based error estimators and material force technique are used to estimate the error on quadtree meshes. The performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary-value problems. (C) 2007 Elsevier B.V. All rights reserved.