  Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$
of all subcontinua of $X$ containing $p$. Given a family of continua
$\mathcal{C}$, a continuum $X\in\mathcal{C}$ and $p\in X$, we say that $(X,p)$
has unique hyperspace $C(p,X)$ relative to $\mathcal{C}$ if for each
$Y\in\mathcal{C}$ and $q\in Y$ such that $C(p,X)$ and $C(q,Y)$ are
homeomorphic, then there is an homeomorphism between $X$ and $Y$ sending $p$ to
$q$. In this paper we study some topological and geometric properties about the
structure of $C(p,X)$ when $X$ is a tree, being the main result that $(X,p)$
has unique hyperspace $C(p,X)$ relative to the class of trees.

General Topology