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The paper studies a PDE model for the growth of a tree stem or a vine, havingthe form of a differential inclusion with state constraints. The equationsdescribe the elongation due to cell growth, and the response to gravity and toexternal obstacles.  The main theorem shows that the evolution problem is well posed, until aspecific "breakdown configuration" is reached. A formula is proved,characterizing the reaction produced by unilateral constraints. At a.e. time t,this is determined by the minimization of an elastic energy functional undersuitable constraints.

Well-posedness of a Model for the Growth of Tree Stems and Vines