Local dendrites with unique hyperspace C(X)

For a continuum X we denote by C(X) the hyperspace of subcontinua of X, metrized by the Hausdorff metric. Let D be the class of dendrites whose set of end points is closed and let LD be the class of local dendrites X such that every point of X has a neighborhood which is in D. In this paper we study the structure of the classes D and LD. As an application, we show that if X∈LD is different from an arc and a simple closed curve, and Y is a continuum such that the hyperspaces C(X) and C(Y) are homeomorphic, then X is homeomorphic to Y.