Treelike quintet systems

Let $X$ be a finite set. We give criterion to say if a system of trees ${\cal P}={T_i}_i$withleafsets$L(T_i) \in {X \choose 5}$ can be amalgamated into a supertree, that is, if there exists a tree $T$ with $L(T)=X$ such that $T$ restricted to $L(T_i)$ is equal to $T_i$.