Is this optimum travelling problem under deadlines NP-hard on trees?
For this to be true, we have to make some assumptions. See comments below

For the tree case, I believe that there is a polynomial time recursive dynamic programming algorithm parameterized by the maximum deadline time. The sub-problems are: if we enter a sub-tree by time $t_a$ and exit the sub-tree by time $t_b$, what is the maximum number of tasks we can finish in the sub-tree? The base cases at the leaves are easy and we memoize from the bottom up. 

If we truly parameterized by the maximum deadline time, then the algorithm would not really be polynomial in the size of the tree. However, the length of the longest path that visits every node in the tree is only polynomial in $|V|$, and we never need to check deadline times later than that.