Check whether a directed, rooted spanning tree is actually some shortest-paths tree in $O(V + E)$ time
Hint: 


This question is telling us to find an algorithm in $O(V+E)$ which immediately suggests us some kind of search in the graph, i.e, DFS.
Using $\pi$ we should be able to build a candidate tree $T$ rooted in $s$ to test if it's shortest-path tree.
In each shortest-path tree, we have that for every $v \in V(T)$, $v \neq s$, the equality $d(\pi(v)) + w(\pi(v), v) = d(v)$ holds.